In a large bowl, combine the bread flour, fine sea salt, sugar, and active dry yeast. Stir briefly to distribute all four dry components evenly before any liquid is added. The yeast and salt require even distribution at this stage — concentrated contact between salt and yeast before the liquid is introduced can suppress yeast activity at the point of contact. A brief 30-second stir ensures they are dispersed evenly through the flour. Bread flour rather than all-purpose is specified for these buns because its higher protein content produces a stronger gluten network capable of supporting the weight of the enrichments — butter and egg — that are about to be incorporated. All-purpose flour produces a softer, less structured bun that does not hold its rounded shape as well during proofing and baking.

In a separate bowl or jug, whisk together the warm milk, warm water, and egg until fully combined — the egg should be fully incorporated into the liquid, not floating as separate streaks. Both the milk and water should be warm to the touch at approximately 35°C: warm enough to activate the yeast but not hot enough to damage it. Pour the liquid mixture over the dry ingredients. Add the softened butter — at room temperature, not melted — at the same time. Melted butter added to the dough at this stage incorporates differently from softened butter: melted butter is entirely liquid and coats the flour particles before the gluten can fully develop, weakening the structure. Softened butter is pliable but still holds its shape, allowing it to be worked progressively into the gluten network during kneading rather than coating everything before development begins. Mix with your hand or a dough scraper until a rough, cohesive dough forms with no dry flour remaining at the bottom of the bowl.

Transfer the rough dough to a clean work surface and knead for 8–10 minutes by hand, or use a stand mixer fitted with a dough hook at medium-low speed for 8 minutes. The dough will feel noticeably sticky at the beginning of the kneading process — more so than a lean dough without butter and egg — and this stickiness is correct and expected for a brioche-style enriched dough. The fat from the butter and the lecithin from the egg yolk both coat the developing gluten strands and increase surface tackiness during early kneading. Resist the urge to add flour: adding flour at this stage changes the recipe's hydration balance and produces a denser, less tender bun. Work through the stickiness — as the gluten network develops and the butter is fully incorporated over 8–10 minutes of kneading, the dough will progressively tighten, become smoother, and pull away from the work surface more cleanly. The finished dough should feel smooth, elastic, and slightly glossy — supple and soft without being sticky when pressed. It should stretch a few centimetres when pulled without tearing.

Shape the kneaded dough into a smooth ball and transfer to a lightly greased bowl. Turn once to coat all surfaces in a thin film of oil, which prevents the surface from drying out during the rise. Cover with plastic wrap or a damp kitchen towel. Allow to rise at room temperature — ideally 24–26°C — for 1–2 hours until the dough has clearly doubled in size. The enrichment from the butter, egg, and sugar all affect the fermentation rate: the fat and egg moderately slow the yeast's activity compared to a lean dough, while the sugar provides additional fuel that accelerates it. The net effect is a rise time similar to a lean dough at 1–2 hours but producing a more flavourful, more complex crumb due to the enrichment's interaction with the developing gluten and yeast activity.

Turn the risen dough out onto a lightly floured work surface. Divide into 4 equal portions — weigh them for consistency, each approximately 105g. Consistent weight produces buns that bake evenly in the same time rather than some over-baking while others remain slightly under. Shape each portion into a tight, smooth ball: cup your hand over the dough piece and roll in a tight circular motion on the work surface, applying gentle downward pressure that tucks the edges underneath and builds surface tension on the top. The smoothness and tightness of the ball's surface determines the evenness of the bun's shape after proofing and baking — a rough, loose ball produces an uneven, asymmetric bun; a tight, smooth ball produces a round, even one. Place the shaped balls on a parchment-lined baking sheet with sufficient space between each for expansion — at least 6–7cm between each ball. Gently flatten each ball with the palm of your hand to create a wide, slightly flat disc shape approximately 10–11cm in diameter. This flattening is important: an unflattened ball proofs and bakes into a dome shape rather than the wide, flat burger bun shape. The slight flattening guides the expansion outward rather than upward during proofing.

Cover the shaped buns loosely with plastic wrap or a clean cloth — lightly greased plastic wrap is preferable as it can be laid over the buns without adhering to their surface. Allow to proof at room temperature for 45–60 minutes until visibly puffy, noticeably expanded, and soft and airy to a very gentle press. The second proof is the stage that most directly determines the softness and airiness of the finished bun's crumb. During this period the yeast continues producing CO2 that inflates the gluten network, and the gluten relaxes into the wider, more rounded shape that the baking will set permanently. A bun that is under-proofed at this stage will have a denser, slightly tight crumb and will spring upward aggressively in the oven rather than expanding evenly in all directions. A bun that is significantly over-proofed will collapse slightly in the oven as the weakened gluten network cannot support the gas structure. The correct endpoint is puffy and light-feeling, with a gentle press leaving an indentation that springs back slowly.

Preheat the oven to 190°C (375°F) in the final 20 minutes of the second proof. In a small bowl, whisk the egg with a small splash of milk until completely smooth — no streaks of unincorporated egg white should remain, as these will produce uneven colour on the bun surface. Using a pastry brush, apply a thin, even coat of egg wash to the top and sides of each proofed bun. The egg wash is what produces the characteristic deep golden, glossy surface of brioche-style bread — the egg proteins and sugars caramelise during baking to create a Maillard-reaction browning that is richer and more appealing than uncoated bread produces at the same temperature. Apply with light, even strokes without pressing down on the buns — pressing deflates the proofed gas structure. Bake for 16–18 minutes until the tops are evenly, deeply golden brown and the buns sound slightly hollow when tapped on the base. Pale buns are under-baked and their crumb will be slightly gummy.

Transfer the baked buns to a wire rack immediately. For extra softness — the technique that produces the characteristically yielding, pillowy surface of the best brioche buns — brush the top surface of each bun lightly with melted butter while still warm, within the first 2 minutes of removing from the oven. The warm bun surface absorbs the melted butter, which adds richness, prevents the crust from firming and toughening as it cools, and produces the soft, slightly shiny, almost melt-in-the-mouth surface that distinguishes a truly excellent brioche bun from a merely good one. Allow to cool for at least 20 minutes before slicing — cutting a hot enriched bread compresses the still-setting crumb and produces a gummy interior.