Preheat oven to 400F
In a large stockpot, cook pasta until almost done. 5 minutes before it is al dente, put broccoli into the pot and cook for the 5 minutes remaining. Drain and set aside.
In the same pot, heat the butter or olive oil and sautee onions until they start to brown.
Add sliced mushrooms and minced garlic and sautee for 5 more minutes.
Sprinkle flour over onion/mushroom mixture and cook for one minute stirring occasionally. Add stock, dissolving any clumps that form.
Add in the milk, Dijon mustard, salt and pepper. Cook sauce until it starts to simmer. Remove from heat.
Stir in one cup of shredded Cheddar cheese and adjust salt and pepper if needed.
In a 9x13 inch baking dish, combine pasta, broccoli, sauce, and chicken. Toss until well combined.
Bake, uncovered for 15 minutes. Add remaining Cheddar cheese and bake for 10 more minutes, uncovered.