Whisk the flour, baking powder, and salt together in a large bowl or use a food processor. Add the cold butter and cut into the dry ingredients with a pastry cutter or pulse several times in the processor until coarse crumbs form. Add 1 cup of milk, then stir/pulse until the dough comes together. Shape biscuit dough into 8 or 9 discs, refrigerate until ready to use.

Melt the butter in a large skillet over medium heat. Add the onion, carrots, celery, mushrooms, and garlic. Cook until softened, about 5 minutes. Stir in flour, salt, pepper, and thyme until flour absorbs all liquid. Stir in broth and ½ cup milk. Simmer until thickened, 7-9 minutes. Stir in mixed vegetables and parsley, remove from heat, cool for 5 minutes.

Preheat oven to 400°F (204°C). Pour filling into greased baking dish. Arrange biscuits on top, brushing with remaining milk. Bake for 25 minutes, then increase oven to 425°F (218°C) and bake for another 5-6 minutes until biscuits are golden brown.

Remove from the oven and allow to cool for 5 minutes before serving. Leftovers can be stored in an airtight container for up to 5 days.