Season chicken to taste on both sides with salt and pepper. In a large skillet over medium-high heat, heat oil until shimmering. Add chicken and cook, without moving, until browned on one side, about 2 to 3 minutes.
Flip and continue to cook until browned on the other side, about 2 to 3 minutes longer. Reduce heat, add 2 tablespoons water to the skillet, and cover.
Cook until the chicken registers 165 degrees on an internal thermometer, about 4 to 5 minutes longer. Remove from pan and let rest for 10 minutes. Slice thinly.
Meanwhile, add milk, broth, and garlic to skillet and bring to simmer. Add fettuccine and stir frequently for 3 minutes. Cook until al dente, about 9 minutes longer.
Stir in heavy cream and parmesan until smooth. Simmer until sauce thickens, about 1 to 2 minutes longer. Season to taste with salt and pepper, then remove from heat and stir in sliced chicken. Garnish with parsley.
Chicken breast: If your chicken breasts are extremely thick, slice them in half lengthwise to ensure the chicken is cooked through in the allotted time.
Fettuccine: Or substitute another long pasta such as linguine or spaghetti.
Yield: This recipe makes 4 servings of Chicken Alfredo.
Storage: Store leftovers covered in the refrigerator for up to 4 days.
Make ahead: The chicken can be cooked and sliced 1 day in advance.