Preheat your grill to medium-high heat. While it's heating, season the chicken breasts with olive oil, salt, and pepper.
Once the grill is ready, place the chicken on it and cook for about 6-7 minutes on each side, or until the internal temperature reaches 165°F (75°C). Remove from the grill and let rest for a few minutes before slicing.
While the chicken is grilling, bring a large pot of salted water to a boil. Add the fettuccine and cook according to package instructions until al dente, usually around 10-12 minutes. Drain and set aside, reserving about 1 cup of pasta water.
In a large skillet over medium heat, melt the butter. Add the minced garlic and sauté for about 1 minute until fragrant, being careful not to burn it.
Pour in the heavy cream and bring it to a gentle simmer. Reduce the heat to low and stir in the grated Parmesan cheese until melted and smooth.
Add the nutmeg, and season the sauce with salt and pepper to taste. If the sauce is too thick, gradually add some reserved pasta water until you reach your desired consistency.
Toss the cooked fettuccine in the Alfredo sauce until well coated. Slice the grilled chicken into strips and place on top of the pasta.
Garnish with chopped parsley before serving.