In a large pot, bring salted water to a boil. Add the spaghetti and cook according to package instructions until al dente. Drain and set aside.
Season the chicken breasts with salt and pepper on both sides. Place the flour in a shallow dish and dredge each breast in the flour, shaking off the excess.
In a large skillet over medium-high heat, heat the olive oil and 2 tablespoons of butter. Add the chicken breasts and cook for about 5-7 minutes on each side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the chicken broth, lemon juice, and capers, scraping any browned bits from the bottom of the pan. Bring the mixture to a simmer and let it cook for 2-3 minutes, allowing the flavors to meld.
Slice the cooked chicken into strips and return it to the skillet. Add the drained pasta to the sauce, and toss to combine. Cook for an additional 2-3 minutes to heat through and allow the pasta to soak up the sauce.
Divide the Chicken Piccata Pasta among plates. Garnish with the remaining 2 tablespoons of butter, chopped parsley, and grated Parmesan cheese if desired.