In a large bowl, combine the sourdough starter, warm milk, and sugar. Stir until the sugar dissolves.
Add the softened butter, flour, salt, and egg. Mix until the dough begins to come together.
Knead the dough for about 5-7 minutes (or use a stand mixer with a dough hook). If using yeast, add it in with the dry ingredients.
Cover the dough with a clean towel and let it rise for 1-2 hours, or until doubled in size.
Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
Once the dough has risen, punch it down and roll it out on a lightly floured surface to about 1/2-inch thickness.
Cut the dough into squares or circles, place a spoonful of filling in the center, and fold the edges around the filling to create a pocket.
Place the kolaches on the prepared baking sheet. Let them rise for an additional 30 minutes.
Brush the kolaches with an egg wash (beaten egg) and bake for 18-20 minutes, or until golden brown.
After baking, brush with melted butter for a soft, shiny finish. Let them cool slightly before serving.
