Position a rack in the lower third of the oven and preheat to 425 degrees . In a medium bowl, melt 1 tablespoon butter in the microwave; stir in the peas and carrots and mushrooms; season with salt and pepper. Press the biscuit dough evenly into a greased 9- to 10-inch cast-iron skillet. Spoon the vegetables on top, reserving the bowl, then bake for 20 minutes.
Meanwhile, in a large nonstick skillet, melt 1 tablespoon butter over medium heat. Add the onion and cook, stirring, until softened, about 7 minutes. Transfer to the reserved bowl along with the chicken and half of the cheese.
Wipe out the skillet, add the remaining 2 tablespoons butter and melt over medium heat. Stir in the flour and cook for 1 minute. Stir in the milk and bring to a simmer, stirring continuously; stir in the remaining cheese and season with salt and pepper. Cover to keep warm.
Preheat the broiler. Spoon the chicken mixture over the baked pizza and broil until golden and bubbly, about 5 minutes. Drizzle with the cheese sauce and sprinkle with the parsley.