Warm milk to around 37°C (not cold, but not too warm). Add milk and sugar to the bowl of a stand mixer and mix until sugar dissolves.

Add sourdough starter, flour, and eggs to the bowl. Mix into a rough dough and let it sit for about 30 minutes.

Add the butter and salt, then knead with a stand mixer until the dough becomes very soft, elastic, and silky. This can take anywhere from 3 to 20 minutes depending on your mixer.

Once dough is ready and pulling away from the sides of the bowl, transfer it to a large bowl for bulk fermentation. Cover with plastic wrap.

Let the dough rise until it's about 50% larger. Turn it out onto a counter and divide into equally sized balls for shaping.

Shape into a loaf or individual rolls. Allow to proof until puffy—this may take time due to the enriched dough.

Preheat oven to 180°C/350°F. Mix an egg with 1 tablespoon of water and brush dough with the egg wash.

Bake for around 30 minutes. If the top browns too quickly, cover with foil and reduce oven to 160°C.

Remove from oven and brush generously with butter while still hot. Cool on a wire rack before slicing.