In a high sided skillet, melt butter on medium heat. Toss in garlic and cook until fragrant, then pour in chicken broth. Bring to a simmer, then turn down heat to medium-low and pour in cream. Allow to simmer and thicken for about 5-10 minutes, keep an eye on it.
After the cream has thickened, turn the heat to low and throw in cheese and whisk to help it melt into the cream. Season with salt and pepper, then splash in a little bit of lemon juice to taste. Whisk really well.
If your sauce isn't as thick as you like (some people like it thin, some thick) make a cornstarch slurry and whisk in.
Mix your chicken, broccoli, and pasta into the sauce. Make sure to coat every piece!