Preheat an oven to 375 F. Coat the inside of a 9 x 13 baking dish with nonstick spray and set aside.
Cooking Spray
In a large saucepan, melt the butter over medium high heat. Add the carrot, celery and onion. Sauté until translucent and cooked through, about 8 to 10 minutes.
2 Tablespoons unsalted butter, 1 carrot, 2 stalks celery, 1 onion
Stir the flour into the vegetables and continue to cook for another 2 minutes. Remove from the heat.
¼ cup all purpose flour
In a large bowl, mix together the vegetables, soup, milk, chicken, salt and pepper. Add to the oiled baking dish.
2 cans, ½ cup milk, 3 to 4 cups shredded, 2 teaspoons kosher salt, 1 teaspoon black pepper
Sprinkle half of the shredded cheese over the top. Arrange the tater tots over the top as evenly as possible. Sprinkle with the remaining cheese.
½ cup cheddar cheese, 1 package
Bake until heated throughout and the top is golden brown, 30 to 35 minutes. The filling should bubble around the edges of the dish. It may be necessary to cover the top with aluminum foil for a portion of the baking time if the top is getting too brown.