Cook pasta according to package directions. Save 1 cup of pasta water. Drain and set aside.
Salt and pepper both sides of the chicken breasts. Heat 2 tablespoons of butter in a large skillet over medium-high heat. Add chicken breasts and cook until deep golden brown on both sides and done in the middle. Remove from the skillet, and set aside.
Add additional 2 tablespoons of butter to the skillet, followed by the minced garlic. Stir the garlic around the pan to avoid burning, and cook for 1 minute. Pour in wine, then let it bubble up and reduce for 1 1/2 to 2 minutes.
Add half-and-half, cream, and extra salt and pepper, whisking constantly until it's all combined. Allow liquid to heat up and thicken for a few minutes. If it gets too thick, you may thin it with a little pasta water that you saved.
Add Parmesan to the pan and stir into the sauce for no more than 1 minute. Remove from heat, then throw the hot pasta right on top of it. Toss it a bit.
Cut the chicken into strips and add to pasta dish and continue tossing until it's all combined. Again, if it gets too gloopy, splash in a little more pasta water.
Taste it, adjust seasonings, and top with minced parsley and extra Parmesan. Serve immediately!