Season the chicken with salt and pepper and sprinkle with 1 tablespoon flour.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat until very hot; cook the cutlets until opaque throughout, 1 to 2 minutes per side. Transfer the cutlets to a plate. (Reserve the skillet.) Reduce the heat under the skillet to medium.
Add the remaining teaspoon oil and garlic to the same skillet; cook until fragrant, about 1 minute. Whisk together the broth, lemon juice and 1 teaspoon flour, and add to the pan. Simmer sauce until reduced and thickened, 2 to 3 minutes. Stir in the vinegar, parsley and capers. Swirl in the butter until sauce is shiny and slightly thickened.
Serve chicken with the sauce. Toss the arugula and freshly grated lemon zest with a light drizzle of olive oil; season to taste with salt and pepper, scatter over chicken. Serve with pasta or potatoes as desired.
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