Preheat the oven to 400 degrees F.
To prepare the filling, melt 2 teaspoons of the butter in a large soup pot over medium-high heat. Add the carrots, onions, celery, turkey, and thyme; cook until the vegetables are soft, about 8 minutes. Add the broth and potatoes; bring to a boil. Cover and simmer 10 minutes.
Combine the flour and water in a small bowl; stir with a whisk until smooth. Return the potato mixture to a boil. Slowly drizzle the flour mixture into the potato mixture, stirring constantly. Cook until the mixture thickens, 1 minute.
Add the peas, salt, and pepper. Spoon the chicken mixture into a large ramekin or ceramic baking dish coated with cooking spray. Place the dough on top of chicken mixture, pressing to the edge of the dish. Cut 2 slits in the top of the crust to allow steam to escape.
Bake until golden and bubbly, 50 minutes.
Let stand 10 minutes before serving.