Combine 1/4 cup broth and miso, stirring well with a whisk until miso dissolves. Stir in 1/2 cup broth. Set aside.
Place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/4-inch thickness using a meat mallet or rolling pin. Place flour in a shallow dish; dredge chicken in flour.
Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 4 minutes on each side or until browned. Remove chicken from pan. Add wine to pan, scraping pan to loosen browned bits. Reduce heat to medium; stir in miso mixture and garlic. Return chicken to pan; cook 3 minutes or until done. Remove from heat; stir in juice, capers, and pepper. Serve immediately.