Make the filling: In a bowl, mix chicken, rice, beans, corn, cheese, cumin, and chili powder.
Assemble burritos: Spoon filling into tortillas, roll tightly, and place seam-side down in a greased baking dish.
Make the sauce: In a saucepan, melt butter, whisk in flour to form a roux, then slowly add milk and broth. Stir until thick, then add cheese and green chilies. Mix until smooth and melty.
Bake: Pour sauce over the burritos, top with extra cheese if you like, and bake at 190°C (375°F) for 20–25 minutes, until bubbly and golden.
