Bring a large pot of salted water to a boil and cook the fettuccine according to package directions until al dente; drain in a colander and toss with a little olive oil to prevent sticking if desired.
While the pasta cooks, heat 2 tablespoons olive oil and 1/4 cup of the butter in a large skillet over high heat.
Season the chicken breasts with 1 teaspoon of the salt and 3/4 teaspoon of the pepper (reserve the remainder). Add the chicken to the hot skillet and cook until golden on both sides.
Cover the pan, reduce heat to medium, and cook the chicken about 5 minutes more or until no longer pink in the center; transfer to a plate, tent with foil, and let rest.
Wipe the skillet if needed, then melt the remaining 1/2 cup butter over medium heat. Add the finely chopped garlic and cook, stirring, about 30 seconds until fragrant.
Reduce heat to low, add the flour and stir to form a paste, then very slowly pour in the heavy cream while stirring constantly until the sauce is smooth.
Cook the sauce over low to medium-low heat, stirring, until steaming and thickened. Stir in 1 cup grated Parmesan until melted. Season to taste with the remaining salt and pepper.
Slice the rested chicken into strips. Divide the drained pasta among plates or bowls, top with chicken, and spoon the Alfredo sauce over everything.
Garnish with chopped fresh parsley and additional Parmesan if desired, then serve immediately.
