In a large sauce pan cook the pasta according to the package to al dente. Drain and set aside. Preheat oven to 370 degrees.
Drizzle the chicken with olive oil, salt and pepper both sides and grill for 5 minutes each side until the center is white. Remove and chop into cubes and set aside.
In a large sauce pan melt butter on medium low and add the flour, stir in a figure 8 for about 5 minutes until it browns and becomes a thick paste (this is a roux) then add the milk and garlic clove, stir often so it does not burn. Cook the sauce on medium low sauce until it just coats the back of your spoon, about 20 minutes. Remove 1/3 of the sauce to a bowl and set aside.
Add the pasta, chicken, broccoli and the pepper to the Alfredo sauce and mix well so every piece of pasta is coated. Turn off heat and move to a 9 x 13 casserole dish. Top with the remainder of the Alfredo sauce and the cheese and place in the oven for 5-10 minutes until the cheese is melted.