Bring a large pot of salted water to a boil. Add fettuccine and cook until al dente, about 8 minutes. Reserve 1 cup of pasta water before draining.
Dab chicken breasts with paper towels to remove moisture. Season with salt and pepper. Heat 2 tbsp butter in a skillet over medium-high heat until foaming. Add chicken and cook until no longer pink, turning once, about 6-7 minutes per side. Transfer to a plate when done.
In the same skillet, add remaining 2 tbsp butter. Add minced garlic and sauté until fragrant, about 1 minute. Gradually pour in heavy cream, whisking to blend. Bring to a gentle simmer, avoiding boiling.
Stir in grated Parmesan cheese and black pepper until the sauce thickens and coats the pan, about 3 minutes. Add reserved pasta water gradually to thin the sauce to desired consistency.
Toss the cooked, drained fettuccine into the sauce until fully coated. Return cooked chicken to the pan and mix gently with the pasta. Serve immediately, garnished with additional Parmesan and fresh pepper if desired.