Make crust: Place flours, thyme, salt, baking powder, and pepper in a food processor; pulse until combined. Add butter; pulse until mixture resembles coarse sand. Drizzle 2 to 3 tablespoons ice water over top of mixture; pulse until mixture begins to clump, 2 or 3 times. Transfer mixture to a work surface; knead until dough comes together, 2 or 3 times. Shape dough into a 5-inch disk and wrap with plastic wrap; chill for 30 minutes.
Make filling: Preheat oven to 400°F. Heat oil in a Dutch oven over medium-high heat. Add carrots, onion, celery, and mushrooms to Dutch oven; cook, stirring occasionally, until mushrooms are tender and other vegetables are tender-crisp, 10 to 12 minutes. Add garlic and thyme; cook, stirring often, for 1 minute. Add turnips, cauliflower, and 21/2 cups broth to Dutch oven; bring to a boil. Whisk together flour and remaining 1 cup broth in a small bowl. Gradually stir flour mixture into broth mixture. Cook, stirring constantly, for 1 minute. Reduce heat to low and simmer, stirring occasionally, until vegetables are tender and mixture has thickened, 7 to 8 minutes. Remove Dutch oven from heat; stir in peas, salt, and pepper. Cover and keep warm.
Place chilled dough on a lightly floured surface. Roll out to a 12-by-8-inch rectangle. Coat an 11-by-7-inch (2-quart) glass or ceramic baking dish with cooking spray; transfer filling to baking dish. Place dish on a rimmed baking sheet lined with aluminum foil. Place dough on filling; fold edges under and crimp. Lightly coat dough with cooking spray. Make 4 to 5 slits in dough to allow steam to escape. Bake until bubbly and golden brown, 25 to 30 minutes.