1 medium onion, coarsely chopped
4 large carrots, cut into 1/2-inch pieces
2 celery ribs, cut into 1/2-inch pieces
1 teaspoon chopped thyme
3 tablespoons unsalted butter
1/2 pound mushrooms, trimmed and quartered
1/4 cup all-purpose flour
4 cups roast turkey meat, cut into 1/2-inch pieces
1 (10-ounce) package frozen baby peas, thawed
3 1/2 cups chicken or turkey stock
For biscuit crust:
2 cups all-purpose flour
2 teaspoons baking powder
1 teaspoon baking soda
1/2 teaspoon salt
1/2 teaspoon black pepper
1 cup coarsely grated extra-sharp Cheddar
1/4 cup grated Parmigiano-Reggiano
3/4 stick cold unsalted butter, cut into 1/2-inch pieces
1 1/4 cups well-shaken buttermilk
Preheat oven to 400°F with rack in middle. Cook onion, carrots, celery and thyme in butter with 1/2 teaspoon salt and 1/4 teaspoon pepper in a 12-inch-wide shallow pot (3- to 4-quart), covered, over medium heat, stirring occasionally, until vegetables are almost tender, 10 to 12 minutes. Add mushrooms and cook, uncovered, stirring, until tender, 5 to 7 minutes.
Sprinkle with flour and cook, stirring constantly, 2 minutes. Stir in stock (3 1/2 cups), scraping up any brown bits, and bring to a boil, stirring, then simmer until slightly thickened, about 3 minutes. Stir in turkey, peas, and salt and pepper to taste. Reheat over low heat just before topping with biscuit crust.
Make biscuit crust and bake pie: Sift together flour, baking powder, baking soda, salt, and pepper into a medium bowl. Add cheeses and toss to coat. Blend in butter with a pastry blender or your fingertips until mixture resembles coarse meal. Add buttermilk and stir just until a dough forms. Drop biscuit dough onto filling in 8 large mounds, leaving spaces between biscuits. Bake until biscuits are puffed and golden brown and filling is bubbling, 35 to 40 minutes. Let stand 10 minutes before serving.