Beat eggs, Parmesan cheese, and parsley together in a shallow bowl. Place flour in a separate shallow bowl.
Gently press chicken breast halves into flour to coat and shake off any excess flour. Dip chicken into egg mixture.
Heat oil in a large skillet over medium-high heat. Cook chicken breasts in hot oil until browned, 4 to 6 minutes per side. Remove chicken from skillet and reduce heat to medium.
Stir butter and garlic into hot skillet. Cook and stir until garlic is fragrant and soft, 3 to 5 minutes. Return chicken breasts to the skillet and drizzle lemon juice over the chicken.
Pour sherry into skillet; stir water and chicken bouillon cube into sherry mixture. Simmer until chicken is no longer pink in the center and the juices run clear, 10 to 15 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add escarole, cover, and steam until tender, 2 to 4 minutes. Serve escarole topped with chicken and pan sauce.