Preheat your oven to 400°F. Bring a large pot of salted water to a boil. Add the pasta and cook until just al dente. In the last 2 minutes, toss in the broccoli florets so they soften slightly. Drain and set aside.
Heat 2–3 tablespoons of olive oil in a skillet over medium heat. Add sliced onions and cook until soft and slightly golden. Stir in minced garlic and cook for about a minute until fragrant.
Sprinkle flour over the onion-garlic mix, stirring constantly to make a roux. Slowly whisk in chicken broth, evaporated milk, half and half, Dijon mustard, salt, and pepper. Let it simmer until it thickens, then reduce the heat and stir in 1 cup of grated cheese until smooth.
In a greased 9x13 dish, add cooked pasta, broccoli, and chicken. Pour the cheese sauce over everything, mixing gently so the sauce coats every bite.
Sprinkle the remaining cheese evenly over the top. Bake uncovered for about 20 minutes, or until golden and bubbling. Let it rest for a few minutes before serving.