Preheat the oven to 400°F. Melt 2 tablespoons butter in a large skillet over medium heat. Add the onion, carrots, and celery and cook until the vegetables begin to soften.
Stir in the 3 tablespoons flour and cook for one minute. Slowly whisk in the chicken broth and 1/2 cup milk until the mixture thickens into a gravy.
Add the diced turkey and frozen peas. Season with salt and pepper to taste, then remove the skillet from the heat.
In a mixing bowl, combine the 2 cups all-purpose flour, baking powder, and 1/2 teaspoon salt. Cut in the remaining 1 tablespoon butter until the mixture looks like coarse crumbs, then stir in the 3/4 cup milk to form a soft dough.
Transfer the turkey filling to a baking dish. Drop spoonfuls of biscuit dough over the top, leaving small gaps between them.
Bake for 20 to 25 minutes until the biscuits are golden and the filling is bubbling around the edges.