Make buttermilk by combining milk and vinegar in a small bowl. Stir and let sit until slightly thickened and curdled, 5 to 10 minutes.
Mix 1 cup flour, sourdough starter, and buttermilk together in a large, non-reactive bowl. Cover and let dough sit in a warm place, 8 hours to overnight.
Add 1 cup flour, sugar, baking powder, salt, and baking soda to the dough; knead lightly until well mixed. Turn dough out onto a work surface that has been sprinkled with remaining 1/2 cup flour. Roll or pat dough out to a thickness of about 1/2 inch. Dip a biscuit cutter into melted butter and cut out biscuits. Reserve any remaining butter.
Place biscuits close together in a greased 9x13-inch pan. Cover and let rest for 30 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Melt any remaining butter and brush over biscuits.
Bake in the preheated oven until golden brown, about 30 minutes. Serve warm.