Combine panko, za'atar spice mix, sesame seeds, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a shallow dish. Place eggs and egg white in another shallow dish; lightly beat with a fork. Sprinkle chicken with remaining 1/2 teaspoon salt and remaining 1/4 teaspoon pepper. Dredge 4 breast halves in panko mixture, then egg mixture. Dredge again in panko mixture.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add 4 coated chicken breast halves to pan. Cook 3 minutes on each side or until lightly browned, turning carefully with a spatula. Place on a baking sheet coated with cooking spray. Repeat procedure with remaining chicken, panko mixture, egg mixture, and oil.
Bake at 400F for 10 minutes or until done. Serve with lemon wedges, if desired.