Place chicken between 2 sheets of heavy-duty plastic wrap; flatten to 1/4" thickness using a meat mallet or rolling pin. Spread breadcrumbs in a shallow plate. Beat egg in a shallow bowl.
Sprinkle chicken with salt and 1/4 teaspoon pepper. Dip 1 chicken breast in beaten egg; coat with breadcrumbs. Repeat with remaining chicken. Cook chicken in 1/4 cup hot oil in a large skillet over medium-high heat 4 minutes on each side or until done.
While chicken cooks, make salad. Gently toss together remaining 1 tablespoon oil, remaining 1/4 teaspoon pepper, hearts of palm, and next 4 ingredients. Serve chicken topped with salad. Garnish, if desired.