1. Heat oven to 400 degrees F.
2. Place chicken on a flat work surface and season both sides with the salt and pepper. Lay 2 slices of prosciutto and 2 slices of cheese and a few basil leaves over each. Roll up tightly and secure with a toothpick.
3. In a large ovenproof skillet, heat 2 tablespoons of the olive oil over medium-high heat. Add the chicken and brown on all sides, about 6 minutes total. Transfer skillet to the oven and bake at 400 degrees F for 12 to 15 minutes or until internal temperature reaches 160 degrees F on an instant-read thermometer.
4. Meanwhile, cook fettuccine following package instructions, until tender but still firm. Drain.
5. Remove chicken from skillet and keep warm. In same skillet, add the remaining 2 tablespoons olive oil, butter and garlic. Cook for 1 minute until garlic is fragrant. Add the drained pasta and toss to coat. Turn off heat and toss with half of the cheese.
6. Cut each piece of chicken into 2 slices and serve over the pasta. Garnish with any remaining basil. Serve with the remaining cheese on the side.