Pre-heat the broiler.
Bring a large pot of water to a boil for the pasta. Salt the water and cook the pasta to al dente, according to the package directions (about 10-12 minutes total, depending on the brand). When the pasta has been cooking for a couple of minutes, add the broccoli to the water and finish cooking everything together, 8 minutes more.
Meanwhile, melt the butter in a saucepot over medium heat. Add the garlic and cook for 2-3 minutes. Whisk in the flour and cook for 1 minute, then whisk in the stock and milk. Let the sauce thicken to coat the back of a spoon; stir in 2 cups of cheese and season with salt and pepper.