In a large bowl, combine the all-purpose flour, fine sea salt, sugar, and active dry yeast. Stir briefly for 20–30 seconds to distribute all four dry components evenly through the flour before any liquid is added. Even distribution at this stage prevents localised concentrations of salt in direct contact with the yeast before the liquid dilutes both — an unimportant consideration in a well-mixed final dough, but a practical one when working with dry ingredients before hydration. All-purpose flour is the deliberate choice here rather than bread flour: its lower protein content produces a softer, more yielding crumb with less noticeable chew — the neutral, accommodating texture of a classic hot dog bun that compresses pleasantly around its filling without the bread-forward character that bread flour would introduce. The difference is immediately perceptible in the finished bun: all-purpose produces the familiar soft, slightly pillowy crumb of a classic hot dog bun; bread flour produces a chewier, more assertively structured roll.

In a separate small bowl or jug, combine the warm milk, warm water, and melted butter or neutral oil. The temperature of both liquids should be 24–27°C — warm enough to activate the yeast and ensure the butter stays fluid, but not hot enough to damage the yeast or produce an uneven dough temperature. Mix briefly to combine, then pour the liquid mixture over the dry ingredients. Mix with your hand or a dough scraper, working from the edges inward, until a rough, shaggy dough forms with no dry flour remaining at the bottom of the bowl. The melted fat — butter or oil — is specified at this quantity and in this liquid form because it incorporates instantly and evenly into the dough at first mixing, lubricating the developing gluten strands uniformly rather than requiring progressive kneading-in as softened butter does. At 30g for 250g of flour, the fat provides sufficient softness and flavour without enriching the dough to the point where it becomes sticky and challenging to handle.

Transfer the rough dough to a clean work surface and knead for 6–8 minutes by hand, or use a stand mixer fitted with a dough hook at medium-low speed for 6 minutes. The classic hot dog bun dough is considerably more manageable than enriched brioche dough — the lower fat content, lower overall hydration, and absence of egg produce a dough that is only slightly tacky from the first minute of kneading and progressively smooths without the extended stickiness that brioche requires patience to work through. The finished dough should feel soft and slightly tacky — soft enough to press easily without resistance, with a slight surface stickiness that does not leave significant dough on your hand when pressed. If it genuinely sticks to everything, add a very small dusting of flour and knead it in. If it feels too firm and resists stretching, add a small splash of water. By the end of the kneading period the dough should be completely smooth on its surface, elastic enough to spring back slowly when poked, and extensible enough to stretch a few centimetres between your fingers before tearing.

Shape the kneaded dough into a smooth ball and transfer to a lightly oiled bowl. Turn once to coat all surfaces. Cover with plastic wrap or a damp kitchen towel and allow to rise at room temperature for 60–90 minutes until roughly doubled in size. The classic hot dog bun dough rises more actively than the brioche version at the same temperature — the lower fat content means the yeast's activity is less moderated by surrounding enrichment, producing a faster, more vigorous rise. In a warm kitchen the dough can double in as little as 60 minutes; in a cooler room it may take the full 90 minutes. Use the visual doubled-volume indicator rather than timing alone.

Turn the risen dough out onto a lightly floured surface and divide into 4 equal portions — each approximately 110g. Weigh them for consistency. Shape each portion into a hot dog bun log using the same technique as the brioche version. Working with one piece at a time, gently press and flatten with your palm into a rough rectangle approximately 10–12cm long. Starting from the long edge nearest you, roll the dough away from you firmly and tightly, pressing the roll against the surface as you go to maintain consistent tension throughout the roll — loose rolling produces irregular crumb structure with gaps and air channels; tight, even rolling produces the uniform, consistent crumb that holds together when bitten. When you reach the far edge, press the seam firmly and completely against the dough surface with the heel of your hand to seal it. Run your fingers along the entire seam length to ensure no section is loose — any unsealed area will open during proofing or baking. Once sealed, place the log seam-side down and lightly roll it back and forth with your palms from the centre outward, applying gradual, even pressure toward the ends to extend the bun to approximately 15–18cm in length and to taper the ends very slightly. The diameter should be even throughout — work slowly and consistently to achieve a uniform cylinder rather than a shape that is thick in the centre and thin at the ends.

Place the shaped buns seam-side down on a parchment-lined baking sheet. For softer-sided buns — the more classic diner-style hot dog bun with a soft, pillowy exterior all around — place the buns slightly closer together than you would for crusty bread: approximately 3–4cm apart rather than 6–7cm. When buns proof close together and their sides make gentle contact during baking, the touching surfaces steam rather than crust, producing the characteristically soft, pale sides of a classic packaged-style hot dog bun. For buns with a more defined crust on all surfaces, space them the full 6cm apart. Both are correct — the choice determines the texture of the sides. Cover loosely with lightly greased plastic wrap or a clean cloth. Allow to proof for 30–45 minutes at room temperature until visibly puffy, clearly expanded in both height and width, and soft to a very gentle press. Begin preheating the oven to 190°C (375°F) in the final 15 minutes of the second proof.

The choice of pre-bake finish determines the crust character of the finished bun — a genuine fork in the recipe at this stage. For a softer, more matte crust characteristic of a classic everyday hot dog bun, brush the top and sides of each proofed bun generously with whole milk using a pastry brush. Milk's fat and protein content promotes browning through the Maillard reaction but does not caramelise as intensely as egg, producing a pale golden, slightly soft surface. For a more golden, slightly crisper, and more defined crust, brush with egg wash — one egg whisked with a splash of water until smooth. The egg wash produces a noticeably deeper golden colour and a slightly more defined, slightly shinier surface. If using sesame seeds, apply them immediately after either finish while the surface is still wet — the milk or egg wash provides sufficient adhesion for the seeds to bond during baking. Scatter generously across the top surface from a slight height for even distribution. Bake for 15–18 minutes until lightly to moderately golden brown depending on your chosen finish — milk-brushed buns will be paler than egg-washed ones at the same baking time. Transfer immediately to a wire rack.