Boil the 16 ounces fettucine in well-salted water following the package timing, then drain it. Leave the pasta warm and slightly firm so it can finish absorbing sauce later.

Blot the 2 chicken breasts dry with paper towels, then season both sides generously with salt and black pepper. A dry, evenly seasoned surface will brown more readily in the skillet.

Warm 2 tablespoons olive oil in a 12-inch skillet over medium-high heat, then place in the chicken and leave it untouched for 5 to 7 minutes. When the underside is deeply golden, it should release more easily from the pan.

Turn the chicken, add 2 tablespoons butter, and cook for another 5 to 7 minutes until an instant-read thermometer reaches 165 °F (74 °C). Move the chicken to a plate, rest it for 7 to 10 minutes, then slice and loosely cover with aluminum foil.

Melt the remaining 1/2 cup butter in a clean, large high-sided skillet over medium-low heat, then stir in 1 tablespoon minced garlic for about 1 minute. The garlic should smell fragrant and stay pale rather than brown.

Whisk in 2 tablespoons all-purpose flour and keep whisking for 1 minute, then take the skillet off the heat and slowly whisk in 2 cups heavy cream and 1 cup whole milk. The roux should foam lightly and turn smooth with no dry flour streaks.

Return the skillet to medium-low heat, then stir in 3/4 cup grated Parmesan cheese and 1/8 teaspoon white pepper until the cheese melts. Keep the heat gentle so the sauce becomes glossy instead of bubbling hard around the edges.

Toss the drained fettucine through the sauce, then let it sit briefly before dividing it among plates with the sliced chicken. The pasta will take on more sauce as it rests; finish with 2 tablespoons chopped parsley and pass extra Parmesan at the table.