Bring a large pot of heavily salted water to a boil. Cook linguine according to package directions until al dente. Reserve 1/2 cup of starchy pasta water before draining.

While pasta is cooking, season the pounded chicken cutlets with salt and pepper. Dredge each piece in flour, shaking off all excess.

In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium-high heat until the butter foam subsides.

Sear the chicken in batches for 3 to 4 minutes per side until golden brown and an internal temperature of 165°F (74°C) is reached. Transfer chicken to a warm plate and tent with foil.

Add the white wine to the skillet, scraping the bottom with a wooden spoon to release the fond. Simmer for 2 minutes until the liquid is reduced by half.

Stir in the minced garlic and cook for 30 seconds until fragrant, then add chicken broth, lemon juice, and capers. Simmer for 5 minutes to reduce slightly.

Reduce heat to low and whisk in the remaining 5 tablespoons of cold butter one tablespoon at a time, whisking constantly to create a smooth, thickened emulsion.

Add the drained pasta to the skillet and toss to coat thoroughly, adding the reserved pasta water as needed to reach the desired sauce consistency.

Slice the chicken into strips and serve immediately over the pasta, garnished with fresh chopped parsley and optional lemon slices.