01 - Preheat a grill pan or outdoor grill over medium-high heat.
02 - Rub the chicken breasts with olive oil, salt, and pepper. Grill for 6-7 minutes per side, or until cooked through (internal temperature 165°F). Remove and let rest for 5 minutes, then slice thinly.
03 - Meanwhile, bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions. Add the broccoli florets to the pasta water during the last 3 minutes of cooking. Drain, reserving 1/2 cup of the pasta water.
04 - In a large skillet, melt butter over medium heat. Add garlic and sauté until fragrant, about 1 minute. Pour in the heavy cream, bring to a gentle simmer, then stir in Parmesan cheese and nutmeg. Season to taste with salt and pepper.
05 - Add the drained pasta and broccoli to the skillet with the cream sauce. Toss to thoroughly coat, adding reserved pasta water a little at a time if the sauce is too thick.
06 - Divide among plates, top with the sliced grilled chicken. Garnish with chopped fresh parsley and extra Parmesan cheese, if desired.