Heat the oil in a pan over medium heat. Add the onion and cook for 2–3 minutes until softened but not browned. Add the flour and stir for about 30 seconds so it absorbs the fat and coats the onion evenly. Add the stock, oyster sauce, hoisin, light soy, dark soy, sugar, sesame oil, and white pepper. Stir well and cook until the mixture thickens into a glossy sauce. Add the diced char siu and fold to coat evenly. Cook for another 1–2 minutes until the filling is thick, sticky, and spoonable rather than loose. Transfer to a bowl and cool completely, then chill until cold. The filling must be fully cold and thick before shaping. Warm filling makes the dough hard to seal and can leak during proofing or baking. Cooking the flour into the fat before any liquid is added coats the starch and removes its raw taste, which is directly what allows the sauce to thicken smoothly once the stock joins rather than turning lumpy or thin.

In a large bowl or the bowl of a stand mixer, combine the flour, yeast, sugar, and salt. Add the warm milk and egg. Mix until a shaggy dough forms. Knead for 5–6 minutes until the dough begins to smooth out, then add the softened butter in pieces. Continue kneading for another 8–10 minutes until the dough is smooth, elastic, and soft. It should be slightly tacky but not wet. If kneading by hand, expect the dough to feel messy when the butter goes in, then come back together as you continue working it. Adding the butter gradually, in pieces, rather than all at once, lets the gluten network absorb each addition before the next arrives — dumping it in together would coat too much of the flour in fat at once, actively working against the structure this enriched dough still needs to develop.

Shape the dough into a ball and place it in a lightly greased bowl. Cover and let rise for 1–1½ hours, or until doubled in size. A full first rise matters here. Under-risen dough gives compact buns with a tight crumb instead of the soft fluffy texture char siu bao should have. That full rise gives the yeast genuine time to build the lightness responsible for the bun's eventual soft, fluffy structure — cutting it short would leave the dough without enough developed lift to bake up properly light.

Turn the dough out onto a lightly floured surface and divide into 12 equal pieces, about 73–75g each. Shape each piece into a ball and let them rest, covered, for 10 minutes so they relax and roll out more easily. Flatten each piece into a round about 10–11cm wide, keeping the center slightly thicker than the edges. Place about 35–40g of the cold char siu filling in the center. Bring the edges up around the filling and pinch firmly to seal. Make sure the seam is completely closed. Place each bun seam-side down on a parchment-lined tray, leaving space between them. Do not overfill. Too much filling makes the buns burst or flatten. Keeping the round's center slightly thicker than its edges gives the base of each bun enough dough to actually support the filling's own weight, while the thinner edges gather and seal more easily around it.

Cover the shaped buns lightly and let proof for 45–60 minutes, until puffy and noticeably lighter. They should not be doubled like dinner rolls, but they should look expanded and soft. If baked too early, they will split and stay dense. That specific, checkable target — puffy and lighter, but not fully doubled — reflects real understanding of what this particular dough needs: enough proofing to bake up soft and light, without needing the same full doubling a plainer dinner roll would require.

Preheat the oven to 180°C. Brush the tops gently with beaten egg and sprinkle lightly with sesame seeds if using. Bake for 15–18 minutes, until the buns are glossy, deeply golden, and fully baked through. The tops should take on a rich egg-washed color while the sides stay pale and soft. Brushing gently, rather than firmly, protects the fragile, fully proofed dough from being pressed down and losing some of the trapped gas responsible for its own eventual softness.

Let the buns cool on the tray for 5 minutes, then transfer to a rack. Serve warm or just slightly cooled. The filling should be thick and glossy inside, not runny, and the dough should be soft, fluffy, and light. That brief cooling period lets the filling's own residual heat settle slightly, making the buns easier to handle and eat without the sticky sauce spilling out the moment they're bitten into.