1. Combine juice, vinegar, shallots, 1/8 teaspoon salt, and sugar; let stand 15 minutes.
2. Place chicken between 2 sheets of heavy-duty plastic wrap; pound to 1/2-inch thickness using a meat mallet or small heavy skillet.
3. Combine breadcrumbs and cheese in a shallow dish. Place flour in a shallow dish. Place egg white in a shallow dish. Sprinkle chicken with 1/8 teaspoon salt and 1/8 teaspoon pepper. Dredge chicken in flour; dip in egg white. Dredge in breadcrumb mixture. Place chicken on a wire rack; let stand 5 minutes.
4. Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add chicken; cook 3 minutes. Turn chicken over; cook 2 minutes or until browned and done.
5. Add 2 teaspoons oil and 1/8 teaspoon pepper to shallot mixture; stir with a whisk. Add greens; toss gently. Place 1 chicken breast half and 1 cup salad on each of 2 plates. Serve with lemon wedges.