Pre-heat oven to 400°F.

Cook pasta in a large pot of generously salted boiling water until al dente.  About 2 minutes before the pasta is done, add the broccoli to the pasta water and stir until combined.  This will save you a dish. Drain the pasta and broccoli.

Using a sauté pan melt the butter or oil over medium-high heat.  Add the onion and cook for 3 minutes, stirring occasionally.  Add the garlic and cook for 3 more minutes, stirring occasionally. 

Sprinkle the flour over the onion and garlic mixture and stir until combined.  Cook for an additional 1 minute. Slowly add in the stock, and stir everything together until mostly smooth and clumps are gone.  Add in the milk, Dijon mustard, salt and pepper and stir until fully combined.  Continue to cook the sauce until it reaches a simmer.  Then remove the sauce from heat and stir in 1 cup of the shredded cheese until thoroughly melted and combined.  

Using a 9 x 13-inch baking dish, combine the cooked pasta, broccoli, chicken and cheese sauce.  Toss until combined.  Even out the ingredients in the casserole dish. 

Bake uncovered for about 15 minutes.  Then remove pan from the oven, to sprinkle the remaining cheddar cheese over the top of the casserole, and bake for 10 minutes or until the cheese is nice and melted.

Serve Warm.