Bring a large pot of generously salted water to a boil. Add the fettuccine and cook until al dente, 8 to 10 minutes. Reserve 1/2 cup of the pasta cooking water before draining.

Pat both sides of the chicken breasts completely dry with paper towels. Season with 3/4 tsp kosher salt and 1/4 tsp black pepper.

Heat 2 tbsp neutral oil in a large non-nonstick skillet over medium-high heat until shimmering, 2 to 3 minutes. Add the chicken and cook without moving it for 5 to 7 minutes. Flip, add 1 tbsp butter between the breasts, and cook another 5 to 7 minutes until the internal temperature reaches 165 degrees F.

Transfer chicken to a cutting board, tent loosely with foil, rest for 5 minutes, then slice.

Without wiping the skillet, reduce heat to medium and melt the remaining 6 tbsp butter, scraping up any browned bits. Add the garlic and cook for about 1 minute until fragrant.

Whisk in the heavy cream, Parmesan, nutmeg, remaining 1/4 tsp salt, and 1/4 tsp black pepper. Simmer for 3 to 4 minutes, stirring frequently, until the sauce thickens enough to coat a spoon. Add reserved pasta water a few tablespoons at a time if the sauce is too thick.

Add the drained fettuccine to the sauce and toss with tongs until fully coated. Divide among bowls, top with sliced chicken, and garnish with extra Parmesan, fresh parsley, and black pepper.