Spread 1/4 cup flour in a shallow dish. Pound each chicken breast half into thin cutlets (for 4 cutlets total). Dry cutlets with paper towels and season with salt and pepper. Dredge one side of each cutlet in flour, shaking off excess.
Heat 1 Tablespoon oil in a large skillet over medium-high heat. Once hot, lay 2 cutlets, floured side down, in the skillet. Cook for 3-4 minutes, or until golden brown. Flip cutlets over and continue to cook until no longer pink, about 1 more minute. Transfer cutlets to a plate and tent loosely with foil. Repeat with remaining 1 Tablespoon oil and 2 cutlets.
Add capers and garlic to the skillet. Return to medium heat and cook for about 30 seconds, until fragrant. Stir in remaining 1 teaspoon flour until incorporated. Whisk in broth, wine and lemon zest strips, scraping any browned bits for additional flavor. Bring to a simmer and cook until thickened slightly and reduced to about 1/2 cup, about 10 to 15 minutes.
Return cutlets into the sauce, simmering until heated through and spooning sauce over chicken, about 30 seconds. Transfer chicken to serving plates. Off heat, remove lemon zest and whisk lemon juice and butter into sauce.
Serve to individual plates and spoon sauce over the chicken.
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