Preheat oven to 400 degrees.
In a large skillet, add 2-3 tablespoons olive oil and saute the chicken over medium heat until cooked through, flipping just once. Set aside chicken in a bowl tented with foil.
In the same pan, add a few tablespoons more of olive oil (or butter if you prefer). Add in thinly sliced onions and saute until they begin to soften, about 4 minutes. Add in garlic and stir until fragrant. Then sprinkle over the flour, stirring, and cook for one minute.
Meanwhile, fill a large pot with salted water, bring to a boil and cook noodles to al dente.
In the pan with onions, pour in the chicken stock mixing well until there are no clumps. Add in evaporated milk, half and half, dijon, and salt and pepper. Coninue cooking until it reaches a simmer.
Add in 1 cup of the grated cheese, stirring until melted. Taste and adjust seasonings as needed. In a 9x13 casserole dish, add in the cooked pasta, broccoli florets, and chicken. Toss until evenly combined. Then pour over the sauce, tossing well until everything is coated.
Bake uncovered for 15 minutes, then remove the casserole dish, add the remaining cheese evenly over the top of the casserole and continue cooking for 10 minutes (or until cheese has melted).