Preheat the Oven: Set your oven to 400°F (200°C) and lightly grease a 9x13-inch baking dish.
Prepare the Biscuits: In a large bowl, whisk together flour, baking powder, and salt. Incorporate the chilled butter with a pastry cutter or your fingertips until the blend takes on a texture similar to coarse crumbs. Add 1 cup of milk and stir until just combined. If needed, add the remaining 2 tablespoons of milk. Lightly flour your hands, shape the dough into a ball, and set aside.
Cook the Vegetables: In a large skillet over medium heat, melt the butter. Add the onion, carrots, celery, and mushrooms. Sauté for 5–6 minutes until softened.
Make the Filling: Sprinkle the flour over the sautéed vegetables and stir well to combine. Cook for 1 minute, stirring constantly. Slowly add the vegetable broth and milk while stirring. Add the salt, pepper, and thyme. Bring the mixture to a simmer and cook until thickened, about 5 minutes. Stir in the mixed vegetables and parsley. Remove from heat.
Assemble the Casserole: Pour the vegetable filling into the prepared baking dish. On a lightly floured surface, roll out the biscuit dough to about 1/2 inch thickness. Cut into rounds using a biscuit cutter or glass. Place the biscuits on top of the filling.
Bake: Place the dish in the preheated oven and bake for 20–25 minutes, or until the biscuits are golden brown and the filling is bubbling.
Serve: Let the casserole cool for a few minutes before serving. Enjoy the rich, comforting flavors of this hearty vegetable pot pie casserole!