Preheat your oven to 350°F (175°C).

Cook the elbow macaroni in a large pot of boiling salted water until al dente, about 8-10 minutes. Add the broccoli florets during the last 2 minutes of cooking; drain and set aside.

Sauté the diced chicken in a skillet with 2 tablespoons of melted butter over medium heat for about 5 minutes until fully cooked. Remove from heat and set aside.

Make the cheese sauce by melting the remaining 2 tablespoons of butter in the same skillet. Whisk in the flour until smooth, and gradually add the milk, whisking continuously until thickened, about 3-5 minutes. Stir in the cheddar cheese until melted.

Combine the cooked pasta, broccoli, and chicken with the cheese sauce in a large bowl. Mix well and transfer to a greased baking dish.

Bake for approximately 20 minutes, or until the top is golden and bubbly.