Make the filling:
Combine the onion, carrots, celery, parsnip, thyme, butter, 1/2 teaspoon salt and 1/4 teaspoon pepper in a wide, shallow (3- to 4-quart) pot set over medium heat. Stir until butter melts then cover and cook, stirring occasionally, until vegetables are almost tender, 10 to 12 minutes. Add the mushrooms and cook, uncovered, stirring, until tender, 5 to 7 minutes.
Sprinkle the vegetables with the flour and cook, stirring constantly, for 2 minutes. Stir in the stock, scraping up any brown bits, and bring it to a boil, stirring, then simmer until slightly thickened, about 3 minutes.
Stir in the turkey, peas, and additional salt and pepper to taste.
Transfer the filling to a 13-by-9-inch baking dish. (See Notes.)
Make the biscuit crust:
Preheat the oven to 400ºF.
Sift together the flour, baking powder, baking soda, salt and pepper into a medium bowl. Add the cheeses and toss to coat. Blend in the butter with your fingertips until the mixture resembles wet sand.
Add the buttermilk and stir just until a dough forms. Drop the biscuit dough onto the pot pie filling in 8 large mounds, leaving spaces between biscuits.
Bake until the biscuits are puffed and golden brown and the filling is bubbling, 35 to 40 minutes. Let stand 10 minutes before serving.
Place the baking dish on a sheet tray before baking in the event any filling spills over.
The filling and biscuit dough can be made one day in advance and stored separately. Reheat the filling over low heat just before topping with biscuit dough and baking.
The fastest way to chicken pot pie is to make the most of a store-bought rotisserie chicken.