Step 1

Preheat oven to 400°F with rack 6 inches from heat source. Whisk together flour and 1 cup of the milk in a small bowl until smooth. Bring remaining 3 cups milk to a boil in a large broiler-safe skillet over medium-high, stirring often. Reduce heat to medium, and gradually whisk in flour mixture until smooth. Return to a boil over medium-high. Boil, stirring constantly, until thickened, 2 to 3 minutes.

Step 2

Stir rice and broccoli into milk mixture until combined. Cook over medium-high, stirring occasionally, until broccoli is bright green, about 2 minutes.

Step 3

Add chicken, fontina cheese, cremini mushrooms, salt, and pepper to rice mixture; stir until cheese is melted. Remove from heat. Stir together panko, thyme, and oil in a small bowl; sprinkle evenly over chicken mixture.

Step 4

Bake in preheated oven until casserole is bubbly, about 15 minutes. Increase oven temperature to broil (do not remove skillet from oven). Broil until top is golden brown, 1 to 2 minutes. Remove from oven, and let cool 5 minutes before serving.