Preheat oven to 350°. Spray a 3-quart baking dish with cooking spray. Set aside.
In a large skillet, cook sausage over medium-high heat until browned, about 5 minutes. Remove from pan with a slotted spoon, and set aside. Reserve rendered fat in pan.
Add bell pepper, onion, and chiles to pan. Cook, stirring occasionally, until tender, about 5 minutes. Remove from pan. Set aside.
Melt butter in pan. Add flour; cook 1 minute, stirring constantly. Gradually add milk and broth, stirring constantly. Bring mixture to a boil; reduce heat to medium-low. Simmer, stirring occasionally, until thickened, about 10 minutes. Add 3 cups cheese, stirring until melted. Stir in sour cream, salt, and pepper. Remove from heat.
In a large bowl, stir together cheese mixture, pepper mixture, sausage, and hash browns until well combined. Spoon into prepared dish. Sprinkle with remaining 1 cup cheese.