In a large pot, melt butter over medium high heat. Add onions, celery, and broccoli and cook until softened, about 3-5 minutes. Stir in flour and cook for 1 minute. Gradually whisk in chicken broth, heavy cream, and milk. Add red pepper flakes, salt, and black pepper. Stir in cooked chicken. Simmer until heated through, about 10 minutes.

Preheat oven to 450°F. Roll out pie crust and cut into strips. Bake on a parchment-lined baking sheet for 6-8 minutes until golden brown. Remove and set aside.

Pour the hot filling into a baking dish. Top with shredded cheese and baked pie crust strips. Bake at 375°F for 20 minutes until bubbly and golden.