In the bowl of a stand mixer, combine the active sourdough starter, bread flour, lukewarm milk, granulated sugar, and salt. Add 2 large eggs. Mix with the dough hook on low speed until a shaggy dough forms and no dry flour remains. Cover the bowl with plastic wrap or a kitchen towel and let it rest for 30 minutes (autolyse).

After the autolyse, knead the dough on medium-low speed for 5-7 minutes until it starts to become smooth and elastic. Gradually add the softened butter cubes, one or two at a time, allowing each addition to be fully incorporated before adding more. This process can take 10-15 minutes. Continue kneading until the dough is very smooth, elastic, and passes the windowpane test (you should be able to stretch a small piece thin enough to see light through it without tearing).

Transfer the dough to a lightly oiled clean bowl. Cover tightly with plastic wrap or a damp kitchen towel. Let it bulk ferment at room temperature (around 70-75°F / 21-24°C) for 4-6 hours, or until it has increased by about 50-75% in volume. Perform 2-3 sets of 'stretch and folds' during the first 2 hours of bulk fermentation at 30-minute intervals. Alternatively, for a deeper flavor, cold proof in the refrigerator for 8-12 hours or overnight.

Gently turn the dough out onto a lightly floured surface. Divide the dough into 8 equal pieces (approximately 90-100g each for standard burger buns). A kitchen scale is recommended for accuracy.

Shape each piece into a tight, round ball. To do this, cup your hand over the dough piece and gently move it in a circular motion on the countertop, creating surface tension. Place the shaped buns on two baking sheets lined with parchment paper, leaving ample space between them.

Loosely cover the shaped buns with plastic wrap or a light kitchen towel. Let them proof at room temperature for 2-3 hours, or until they are visibly puffy and have increased in size significantly (the 'poke test' should leave a slight indentation that springs back slowly). If cold proofing, allow them to come to room temperature for about 1 hour before proofing.

About 30 minutes before baking, preheat your oven to 375°F (190°C).

In a small bowl, whisk the extra large egg with 1 tablespoon of milk or water to create an egg wash. Gently brush the tops of the proofed buns with the egg wash. Sprinkle with sesame seeds, if desired.

Bake for 18-22 minutes, rotating the baking sheets halfway through, until the buns are deep golden brown on top and bottom.

Transfer the baked buns to a wire rack to cool completely before slicing and serving. This helps prevent a gummy texture inside.