Bring a large pot of generously salted water to a boil. Add 8 ounces dried fettuccine and cook until al dente, 8 to 10 minutes. Before draining, scoop out 1/2 cup of the pasta cooking water and set it aside. Drain the pasta and set aside.
While the water comes to a boil, pat 2 boneless, skinless chicken breasts completely dry with paper towels. Season all over with 3/4 teaspoon kosher salt and 1/4 teaspoon black pepper.
Heat 2 tablespoons neutral oil in a large skillet (not nonstick) over medium-high heat for 2 to 3 minutes. Swirl the pan right before adding the chicken to coat the bottom with oil evenly.
Add the chicken and cook without moving it until the bottom is deep golden-brown and releases easily from the pan, 5 to 7 minutes. If the chicken feels stuck when you try to flip it, give it another minute.
Flip the chicken and drop 1 tablespoon of European-style unsalted butter between the two pieces. Swirl the pan gently to distribute the melting butter. Continue cooking until the chicken reaches an internal temperature of 165°F, 5 to 7 minutes more.
Transfer the chicken to a cutting board and let it rest for 3 minutes. Slice into 1/2-inch-thick pieces and cover loosely with aluminum foil to keep warm.
Add the remaining 7 tablespoons European-style unsalted butter to the same skillet over medium heat. Once melted, add 2 cloves minced garlic and cook, stirring, for 30 seconds to 1 minute until very fragrant.
Whisk in 1 cup heavy cream, 1 cup finely grated Parmesan cheese, 1/4 teaspoon ground nutmeg, the remaining 1/4 teaspoon kosher salt, and the remaining 1/4 teaspoon black pepper. Simmer for 3 to 4 minutes, whisking occasionally, until the sauce thickens. If the sauce is too thick, add the reserved pasta water a few tablespoons at a time, whisking until the consistency is right.
Add the drained fettuccine to the sauce and toss well to coat. Divide among serving bowls, top with sliced chicken, and garnish with fresh parsley, extra Parmesan, and black pepper as desired. Serve immediately.