Heat the oven to 400°F and grease a 9-by-13-inch baking dish.
Melt the butter in a large skillet and cook the onion, carrots and celery until nearly tender, about 8 minutes.
Stir in the flour for 1 minute, then gradually add the broth and milk and simmer until the sauce thickly coats a spoon.
Add the salt, black pepper, cooked chicken and peas and spread the filling in the baking dish.
Spoon the mashed potatoes over the filling, rough up the surface with a fork and sprinkle on the cheddar.
Bake until the filling bubbles and the potato ridges brown, 25 to 30 minutes.