Cook pasta according to package directions until al dente in generously salted water. Reserve about 1 cup pasta water for later. Drain and set aside.
Season the chicken cutlets with salt and pepper generously (about ¼ tsp salt per cutlet). Dredge each cutlet with flour, shaking off any excess.
Heat a few drizzles of olive oil in a large skillet over medium heat. Cook the chicken on both sides until golden brown and fully cooked through in the center. Remove and set aside.
To the same pan, add 2 tablespoons of butter. Sauté the shallots and garlic for about 1 to 2 minutes until fragrant. Then, add the capers and wine and scrape the brown bits (use a wooden spoon to avoid scratching).
Next, stir in the chicken broth, lemon juice (to taste), and the lemon slices. Season with salt and pepper to taste.
Return the chicken to the pan. Bring to a low simmer, and simmer for 2 to 3 minutes until the sauce is slightly thickened. Then remove the chicken from the pan and transfer to a plate. Leave the sauce in the pan.
Turn off the heat. Then, immediately add 3 tablespoons of butter into the sauce and stir until thickened and glossy. Spoon half of the lemon sauce and garnishes over the chicken. Leave the other half of the sauce in the pan.
Then, toss the cooked pasta with the sauce in the skillet, plus some of the reserved pasta water as needed to achieve the right consistency.
Top the pasta with the chicken and sauce. Garnish with Italian parsley and more lemon slices. Enjoy!