Spray a skillet with cooking spray and place over medium heat; cook chicken in the hot skillet until no longer pink in the center and the juices run clear, 6 to 8 minutes per side. Set chicken aside.
Preheat oven to 350 degrees F (175 degrees C). Spray a 9x13-inch casserole dish with cooking spray.
Place broccoli in the prepared casserole dish.
Melt 3 tablespoons butter in a saucepan over medium heat; stir flour into the melted butter until combined and thickened, 3 to 5 minutes. Slowly pour in chicken stock and milk, stirring constantly; continue cooking and stirring until sauce is thickened, 5 to 10 more minutes.
Stir onion, red bell pepper, serrano chile pepper, habanero pepper, and carrot into the sauce; simmer until vegetables are slightly softened, 5 to 10 minutes.
Cut the cooled chicken into cubes and sprinkle over broccoli. Pour sauce over chicken and broccoli.
Heat 1 teaspoon butter in a skillet over medium heat; cook and stir mushrooms until tender, 5 to 10 minutes. Spoon over sauce layer in the casserole dish. Sprinkle bread crumbs and Parmesan cheese over mushroom layer.
Bake in the preheated oven until cooked through and bubbling, about 30 minutes. Turn on the oven's broiler; broil casserole until topping is browned, 2 to 4 minutes.
Spoon casserole into 4 bowls and sprinkle 1/4 cup arugula atop each serving.