Prep the chicken: Pat the chicken cutlets dry. Season both sides with salt and pepper.
Place flour in a shallow dish and dredge each cutlet, shaking off the excess.
Sear to golden: In a large skillet over medium-high heat, heat the olive oil and 1 tablespoon of butter. Add the chicken in a single layer (work in batches if needed). Cook 3–4 minutes per side, until golden and just cooked through.
Transfer to a plate and tent loosely with foil.
Sauté the aromatics: Reduce heat to medium. Add the remaining 1 tablespoon butter to the pan. Stir in the garlic and cook 30 seconds, just until fragrant.
Avoid browning.
Deglaze: Pour in the white wine (or chicken broth). Use a wooden spoon to scrape up the browned bits. Simmer 2–3 minutes to reduce slightly.
Build the sauce: Add the chicken broth, lemon juice, and lemon zest.
Stir in the capers. Simmer 3–4 minutes, letting the flavors meld.
Make it creamy: Lower the heat and stir in the heavy cream. Let it gently bubble for 2–3 minutes to thicken.
If using Parmesan, whisk it in now until melted and smooth. Taste and adjust salt and pepper.
Return the chicken: Slide the cutlets and any juices back into the pan, turning to coat. Simmer 1–2 minutes, just to warm through and let the sauce cling.
Finish and serve: Sprinkle with chopped parsley.
Serve the chicken with extra spoonfuls of sauce over pasta, rice, or potatoes.