In the bowl of a stand mixer, whisk together the warm whole milk, active sourdough starter (or discard), granulated sugar, and one room-temperature egg until well combined.

Add the bread flour and fine sea salt to the wet ingredients. Attach the dough hook and mix on low speed until a shaggy dough forms and no dry flour remains.

With the mixer running on medium-low, begin adding the softened butter one tablespoon at a time. Wait until each piece is fully incorporated before adding the next.

Once all the butter is added, continue to knead on medium speed for 8-10 minutes. The dough should become incredibly smooth, elastic, and pull away completely from the sides of the bowl (it will clear the bowl).

Form the dough into a smooth ball, place it back in the bowl, cover it with plastic wrap or a damp towel, and let it rise at room temperature for 4 to 6 hours, or until roughly doubled in size. (If using cold discard, this bulk fermentation may take longer).

Turn the doubled dough out onto a clean, un-floured work surface. Divide the dough into 8 equal pieces (about 100g to 110g each).

Shape each piece into a tight, smooth ball by pulling the edges toward the bottom and rolling it under the cage of your hand.

Place the balls on a parchment-lined baking sheet, spacing them well apart. Here is the crucial step: use the palm of your hand to gently but firmly flatten each ball into a disc about 3.5 inches wide. (If you don't flatten them, they will bake into tall dinner rolls instead of burger buns).

Cover the flattened buns loosely with a towel and let them proof for 1 to 2 hours, or until very puffy and light.

Preheat your oven to 375°F (190°C).

In a small bowl, whisk the remaining egg with 1 tablespoon of milk. Gently brush the top of each proofed bun with the egg wash, then sprinkle generously with sesame seeds.

Bake for 15-18 minutes, or until the buns are a deep, glossy golden brown.

Transfer the buns immediately to a wire rack to cool completely before slicing.