Bring a large pot of salted water to a boil. Add the fettuccine and cook until al dente, about 10-12 minutes. Reserve 1 cup of pasta water and drain the rest.
Sprinkle salt and pepper on the chicken breasts. Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large skillet over medium-high heat.
Add the chicken and cook for about 5-7 minutes on each side until the internal temperature reaches 165°F. Remove from the skillet and let rest, then slice into strips.
In the same skillet, melt the remaining tablespoon of butter over medium heat. Add the minced garlic and sauté for approximately 1 minute until it becomes fragrant. Then, sprinkle in the flour and stir continuously for one minute to create a roux.
Gradually whisk in the heavy cream and whole milk until the mixture is smooth. Let the sauce simmer for 3-4 minutes until it begins to thicken, then stir in the grated Parmesan cheese. Adjust the seasoning with salt and pepper as needed.
Combine the drained pasta with the sauce in the skillet. Toss thoroughly to ensure the pasta is evenly coated, adding a bit of reserved pasta water if needed to achieve your preferred consistency.
Arrange the sauced pasta on plates and top with sliced chicken. Garnish with extra Parmesan cheese and a sprinkle of fresh chopped parsley. Serve warm.