Pound chicken breasts to about 1/2-inch thickness. Season both sides of the chicken cutlets with 1/2 tsp salt and 1/4 tsp pepper.

Place flour in a shallow dish. Lightly dredge each chicken cutlet in the flour, shaking off excess.

Heat 2 tbsp olive oil and 1 tbsp butter in a large skillet over medium-high heat.

Once butter is melted and shimmering, add half of the chicken cutlets to the skillet. Cook for 3-4 minutes per side, until golden brown and cooked through. Remove chicken to a plate and set aside.

Add remaining 1 tbsp olive oil and 1 tbsp butter to the skillet, then cook the second batch of chicken. Remove and set aside with the first batch.

While chicken is cooking, bring a large pot of generously salted water to a rolling boil.

Add linguine and cook according to package directions until al dente. Reserve 1 cup of pasta water before draining. Drain the pasta.

Reduce heat to medium. Deglaze the same skillet by pouring in the white wine (or chicken broth) and scraping up any browned bits from the bottom with a wooden spoon. Let it simmer for 1-2 minutes until slightly reduced.

Add chicken broth, fresh lemon juice, and capers to the skillet. Bring to a simmer and cook for 3-5 minutes, allowing the sauce to reduce slightly and flavors to meld.

Season the sauce with remaining 1/2 tsp salt and 1/4 tsp pepper, or to taste.

Remove the skillet from heat. Whisk in the cold butter cubes, one at a time, until fully incorporated and the sauce is slightly thickened and glossy.

Add the cooked and drained pasta directly to the skillet with the piccata sauce. Toss to coat the pasta evenly. If the sauce seems too thick, add a splash of the reserved pasta water to reach desired consistency.

Slice the cooked chicken cutlets into strips or leave whole, and return them to the skillet with the pasta, gently tossing to combine.

Garnish with fresh chopped parsley and optional grated Parmesan cheese. Serve immediately.