Place chicken cutlets between parchment or plastic wrap. Pound lightly with a meat mallet or rolling pin until about ½-inch thick.
Season flour with salt and pepper. Dredge chicken lightly in the flour, shaking off excess.
Sear the chicken
Heat olive oil and 2 tbsp butter in a large skillet over medium-high heat.
Add chicken and cook 3–4 minutes per side until golden and cooked through. Remove to a plate.
In the same skillet, add garlic and sauté 30 seconds.
Deglaze with white wine (or broth), scraping up any browned bits. Let it simmer until reduced by half, about 2–3 minutes.
Add chicken broth, lemon juice, and capers. Simmer another 2 minutes.
Return chicken to the pan and spoon sauce over it. Simmer gently 2 minutes so flavors combine.
Remove from heat and stir in the remaining 1 tbsp butter for a silky finish.
Garnish with fresh parsley and lemon slices.