Fill a large pot with salted water and bring it to a boil. Add the pasta and cook according to the package instructions until al dente.
About 3 minutes before the pasta is done, add the broccoli florets to the boiling water with the pasta.
Drain the pasta and broccoli together and set aside.
While the pasta cooks, season the chicken breasts with salt and pepper on both sides.
In a skillet over medium-high heat, heat olive oil. Add chicken breasts and cook on both sides until cooked to 165°F internally.
Remove chicken from skillet, let it rest briefly, then slice into strips.
In the same skillet over medium heat, melt the butter. Add the minced garlic and cook for about 1 minute until fragrant.
Pour in the heavy cream and bring to a gentle simmer.
Gradually mix in the grated Parmesan cheese until the sauce is creamy and the cheese melts completely.
Add ground nutmeg. Taste and adjust seasoning if needed.
Add the drained pasta and broccoli to the skillet with the Alfredo sauce. Toss to coat.
Add the sliced chicken to the skillet and stir to combine.
Serve the chicken broccoli Alfredo warm, garnished with additional Parmesan cheese if desired.