Chicken Broccoli Alfredo

I'm calling this one a huge success.  Not only did Laura eat dinner (which is rare, to say the least), but Maddy had THREE helpings!  I think that next time, I'll double the alfredo sauce.  You could taste the cheese for sure, but it wasn't really "saucy" at all. Here's the recipe:

3 boneless skinless chicken breasts, seasoned liberally w salt, pepper and garlic
1 lb spaghetti (I used whole wheat)
4 crowns of broccoli
1/4 c butter
1 c heavy cream
1 clove garlic, crushed
1 1/2 c grated cheese (parmesan, gruyere, etc)
1/4 c parsley, chopped

Grill chicken until done, and cube. At the same time, steam the broccoli and drain. Boil pasta to al dente.  Put all in large bowl, and pour alfredo (see below) over top, then toss.

Alfredo:
Melt butter in saucepan over medium-low heat.  Stir in cream, simmer for about five minutes.  Add in the garlic and the cheese.  Whisk together, heating thoroughly to melt cheese. Stir in parsley.