Preheat the oven to 400°F (200°C) and line a baking sheet with parchment or use a nonstick sheet.
Trim and cut the chicken breasts into 1-inch cubes and place them in a large mixing bowl.
Lightly whip the egg white, pour it over the chicken cubes, and toss until each piece is coated.
Add the black pepper, cayenne, onion powder, and salt to the chicken, then add the whole-wheat panko and toss to coat evenly, pressing crumbs onto pieces as needed.
Arrange the coated chicken pieces in a single layer on the prepared baking sheet, spacing them slightly apart. Drizzle with the tablespoon of oil or spray lightly with an oil sprayer.
Bake for about 30 minutes, turning once if desired, until the nuggets are golden and the chicken is cooked through (no pink inside and juices run clear).