Cook pasta according to package instructions in well-salted water. Drain.
Pat chicken breast dry with paper towels. Season generously with salt and pepper.
Heat the olive oil over medium-high heat, in a 12-inch skillet. Cook chicken breasts in the hot skillet until golden brown, about 5-7 minutes without flipping them or moving them in the pan. Turn breasts over and add 2 tablespoon butter. Cook the breast another 5-7 minutes or until an instant thermometer reads 165 degrees. Do not overcook or chicken will be dry. Remove to a plate and allow to rest for 7-10 minutes. Slice chicken and cover with aluminum foil; set aside.
In a clean large, high sided skillet, melt the remaining 1/2 cup butter over medium-low heat and saute garlic until fragrant, about a minute. Sprinkle flour over butter and cook, whisking constantly for 1 minute. Remove pan from heat and slowly whisk in cream and milk until smooth.
Return skillet to medium-low heat and stir in 3/4 cup Parmesan cheese and white pepper and simmer gently until cheese is melted.
Toss cooked pasta with sauce. (At first, the pasta may seem soupy but, as it sits, the pasta will quickly absorb more sauce and you will have plenty of sauce to coat the pasta generously and it won't be dry when you go back for seconds!) Arrange pasta on plates and top with sliced chicken. Sprinkle with parsley and serve extra Parmesan on the side.