Slice or pound chicken breasts to even thickness for quick cooking. Season lightly with salt and pepper. Even thickness prevents dry or undercooked spots.
Place flour, egg mixture, and breadcrumb mixture in separate bowls. Coat chicken first in flour, then egg, then breadcrumbs. Press crumbs gently so they adhere well.
Heat oil in a skillet over medium heat and cook cutlets until golden brown. Cook each side until crisp and fully cooked inside. Drain cooked cutlets on paper towels.
In a small pan, melt butter and stir in capers and lemon juice. Let sauce warm briefly so flavors combine. Remove from heat to prevent burning.
Spoon warm caper butter sauce over crispy cutlets. Sprinkle fresh parsley on top before serving. Serve immediately for best crisp texture.