In a large pot bring salted water to a boil and cook spaghetti according to package directions until al dente. Reserve a cup of pasta water; you'll use it later to adjust sauce. Drain pasta and rinse under cold water to stop cooking. Meanwhile season chicken cutlets with salt and pepper then dredge in flour, shaking off excess. Heat olive oil in a 12-inch skillet over medium heat; cook cutlets until golden brown on both sides and cooked through, about 4 minutes per side. Transfer chicken to a plate.
Return the skillet to medium heat and add two tablespoons of butter. When it melts add shallot and garlic and sauté until fragrant, 1 to 2 minutes. Stir in capers then pour in white wine, scraping any brown bits from the pan. Add chicken broth, lemon juice and lemon slices. Taste and adjust salt or pepper if needed.
Nestle chicken cutlets into the sauce and lower heat. Let it come to a gentle simmer and cook for 2 to 3 minutes, until sauce thickens slightly. Remove chicken to a plate and leave the sauce in the skillet.
Turn off heat and stir in remaining three tablespoons of butter until the sauce looks glossy and thick. Spoon half of the sauce over chicken. Add spaghetti to the skillet and toss with the remaining sauce, adding reserved pasta water as needed to reach the right consistency. Top pasta with chicken, sprinkle parsley and serve.