Heat the butter and milk in a small saucepan on the stove or in a small bowl in the microwave until warm, but no warmer than 110 degrees F. If the liquid is hotter than 120 degrees F, it can kill the sourdough starter, which will prevent the dough from rising.
113g (1/2 c) unsalted butter, 240g (1 c) milk
Meanwhile, add the bubbly sourdough starter, egg, and sugar to the bowl of a stand mixer with a paddle attachment. Mix on low speed until combined, then gradually add the warm milk and melted butter mixture while continuing to mix.
This can also be done by hand.
200g (1 c) active sourdough starter, 1 egg, room temperature, 70g (1/3 c) sugar
To the wet ingredients, add the flour and salt and mix again on low speed. Stop to scrape the sides of the bowl to incorporate all the dry ingredients, and then continue to mix on low for 1-2 minutes.
450g (3 3/4 c) 9g (1 1/2 tsp) salt
The dough will look thick and sticky, like a cake batter. Cover with plastic wrap or what I like to use—a reusable shower cap—and allow the dough to rest for 30 minutes. This allows the gluten to rest before kneading.