Set up a station to prepare your chicken with a clean cutting board, and a wide, shallow bowl of flour. On the cutting board, use a meat mallet to pound the chicken breasts to ¼ inch thickness. Sprinkle the thinned chicken breasts with ½ teaspoon salt and ¼ teaspoon pepper. Dip them into the flour to coat both sides, then shake off the excess flour. Lay them out on a sheet pan and set aside. Get your pasta water boiling so that everything is ready at the same time.
In a large skillet, heat 2 tablespoons of oil over medium heat. Gently place the chicken into the hot oil. Cook for two to three minutes on each side, or until it's no longer pink. You can also use a digital thermometer to test the internal temperature; it should read 165°F. Remove the chicken and keep warm while you finish the pasta. Meanwhile, cook the pasta according to package directions for al dente. Editor's Tip: To tell when your olive oil is ready, touch a wooden stick to the bottom If it sizzles, you're good to go.
In the same pan that you cooked the chicken, heat the remaining oil over medium heat. Add the garlic and stir for 30 to 60 seconds or until lightly browned. Deglaze the pan with wine or chicken stock, and then increase the heat to medium-high. Stir to loosen any browned bits from the pan, and cook until the liquid reduces by half. Stir in the lemon juice.
Strain pasta, but reserve at least 1.5 cup of pasta water. Place the pasta in a large bowl or back in the pan.
Sprinkle some of the grated cheese into the pasta and some of the fresh or dried parsely and stir. Repeat until half gone. Add pasta water to moisten it all up, add rest of cheese and parsely. Add more water as needed.
Add half the garlic mixture Toss everything to combine.
Slice the cooked chicken and place it over the pasta for serving. Drizzle the remaining garlic mixture over it, and then sprinkle the remaining parsley. Serve with additional grated Parmesan and lemon wedges. Place a heat-safe measuring cup in the empty strainer to remind yourself that you want to save some