In a large bowl, combine the cabbage, broccoli, carrots, mayonnaise, relish, and ¼ teaspoon salt.
Place the egg and panko in separate shallow bowls. Dividing evenly, season the chicken with ¼ teaspoon each salt and pepper. Dip in the egg (shaking off any excess), then coat with the panko, pressing gently to help it adhere.
Heat the oil in a large skillet over medium-high heat. Working in batches, cook the chicken until cooked through, 3 to 4 minutes per side. Serve with the slaw.