In a large mixing bowl, whisk together the sourdough starter, warm milk, egg, sugar, vanilla, and salt. Gradually stir in the flour until a sticky dough forms. Knead until smooth, 5–7 minutes (or use a stand mixer). Cover and chill for 1 hour.
While the dough chills, place cold butter between two sheets of parchment. Roll into a 6x6-inch square. Chill again if it gets too soft.
On a lightly floured surface, roll the dough into a 12x12-inch square. Place the butter block in the center like a diamond. Fold each corner of the dough over the butter to completely encase it. Roll into a 10x20-inch rectangle. Fold into thirds like a letter. Wrap and chill for 30 minutes.
Repeat this process two more times, chilling in between.
Roll the dough to 1/4-inch thickness. Cut into 4-inch squares. Add 1 tbsp filling to each center, then fold or twist corners to form shapes.
Arrange on parchment-lined trays. Cover and let rise overnight in the fridge (8–12 hours).
Preheat oven to 400°F. Brush with egg wash. Bake for 18–20 minutes or until golden and puffed. Let cool before glazing.
Optional Glaze:
Drizzle with powdered sugar glaze once pastries are cool.