To Pound or Slice Chicken into Thin Cutlets Cut the breasts into two halves. Pull out the tenders so you can flour and cook them separately. To pound, place each chicken breast half between 2 sheets of heavy-duty plastic wrap or inside a large plastic baggie; pound with a meat mallet, small heavy skillet, or rolling pin, to 1/2-inch thickness. Cut each thin cutlet in half, if you like the idea of 8 small pieces rather than 4 large. Alternatively you can slice the chicken into cutlets: set each breast half on a flat surface and use a very sharp knife to slice it horizontally through the middle* (see link below for tutorial on slicing and pounding chicken into cutlets)
To Flour the Chicken: Mix 1/2 cup flour with 1 teaspoon salt and 1/4 teaspoon pepper and spread seasoned flour onto a shallow dish. Dredge chicken in flour. Shake off excess. Set pieces on a plate as you go.
To Brown the Chicken: Melt 1 tablespoon butter in a large skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add chicken to pan and sauté 2-3 minutes per side or until done. If cooking more than 4 cutlets, add a little more oil and butter before adding the second batch. Transfer chicken to a plate as you go.
Make the Pan Sauce and Finish the Dish: Heat remaining 1 tablespoon oil in pan over medium heat; swirl to coat. Add garlic; sauté 1 minute, stirring constantly. Add wine; bring to a boil, scraping pan to loosen browned bits. Cook until liquid almost evaporates, about 2 minutes, stirring occasionally. Add 1/2 cup of the chicken broth and bring to a boil. Simmer for 3-4 minutes until reduced by about a third. While the sauce is simmering combine the remaining teaspoon of flour with the remaining 1/4 cup of chicken broth in a small cup and stir until smooth. Whisk the flour mixture to the sauce and cook for 1 minute or until slightly thick, stirring frequently. Lower the heat to a simmer; whisk in remaining 1 1/2 tablespoons butter, lemon juice, and capers. Return chicken to the pan. Simmer gently in the sauce for two minutes or until heated through.
To Serve: Plate chicken. Top each serving with sauce. Garnish with a slice of lemon and a sprinkle of parsley.