Place the chicken between sheets of plastic and pound to an even 3/4-inch thickness. Pat dry, season both sides generously with salt and pepper, and let rest 10 minutes.
Stir the flour with a pinch of salt and pepper on a shallow plate. Dredge each chicken breast lightly, shaking off the excess.
Heat a 12-inch skillet over medium heat. Add olive oil and 1 tablespoon of the butter. Sear the chicken until well browned, 4–6 minutes on the first side and 3–4 minutes on the second. Transfer to a plate and tent with foil.
Lower the heat to medium-low. Melt the remaining 2 tablespoons butter in the pan, then add the garlic. Cook, stirring, until fragrant, 45–60 seconds (do not brown).
Deglaze with the chicken broth and 2 tablespoons of the lemon juice, scraping up any browned bits. Stir in the capers and simmer 2–3 minutes to reduce slightly.
Pour in the heavy cream and any resting juices from the chicken. Bring to a gentle simmer over low heat and cook 3–5 minutes, stirring, until the sauce lightly coats a spoon. If it thickens too much, whisk in a splash of broth or water.
Off the heat, stir in the remaining 1/2 tablespoon lemon juice, the parsley, and the optional lemon zest. Taste and season with more salt and pepper as needed.
Nestle the chicken back into the skillet, turning to coat in the sauce. Simmer on low for about 2 minutes to warm through.
Garnish with extra parsley and thin lemon slices. Serve hot over pasta, rice, or creamy mashed potatoes.