Melt one tablespoon of butter in a Dutch oven, over medium high heat. Add garlic and sauté until fragrant, about 30 seconds. Add the onion and broccoli. Sauté until vegetables have started to soften, about 5 minutes. Transfer broccoli mixture to a bowl. Set aside.

Melt the remaining 7 tablespoons butter in the Dutch oven, over medium high heat. Whisk in flour and cook until for 2 minutes, stirring constantly.

Slowly whisk in milk and broth, stirring to combine. Add broccoli mixture back to the milk mixture. Bring to a boil. Reduce heat and simmer until soup has thickened to the consistency of condensed soup, about 10 to 15 minutes, stirring occasionally. Season with salt and pepper, to taste. Remove from heat and set aside.

Preheat oven to 350 F.

Cook the pasta for 3 minutes less than the time listed in the instructions. Drain well.

Meanwhile blanch broccoli and drain well.

Place cooked pasta, broccoli and chicken in a 9 x 13 inch pan. Stir to combine. Pour the cream of broccoli soup evenly over the pasta mixture. Sprinkle 1 1/2 cups of the cheese, salt and pepper over the pasta mixture. Stir to combine. Sprinkle the remaining 1/2 cup of cheese over the top of the pasta mixture.

Bake until contents are heated through, about 25 to 30 minutes.

Serve hot.