Warm the whole milk to about 37°C (98°F) and combine it with sugar in a stand mixer bowl, stirring until the sugar dissolves. Next, add the bubbly sourdough starter, bread flour, and eggs. Mix on low speed until a rough dough forms. Allow the dough to rest for 30 minutes.

Add the softened salted butter and salt to the dough mixture. Knead the dough on medium speed for about 8-10 minutes until it becomes soft, elastic, and slightly sticky.

Transfer your kneaded dough into a greased bowl, covering it with a clean kitchen towel. Let the dough rise at room temperature until it has increased in size by about 50%, approximately 1 to 2 hours.

Turn the dough onto a lightly floured surface. Gently deflate the dough and divide it into equal portions. Shape each piece into a ball and arrange them in a greased bread pan.

Cover the shaped dough again with a towel and let it proof until puffy, usually about 30-60 minutes.

Preheat your oven to 180°C (350°F). Whisk together 1 egg with a tablespoon of water for the egg wash. Brush generously over the top of the loaf, then bake for approximately 30 minutes.

Once baked, brush the loaf with melted butter and allow to cool for a few minutes before transferring to a wire rack.