Preheat oven to 400 degrees F. Use a baking sheet with sides and lightly coat it with cooking spray.

Mix cheese, ham, mustard and pepper in a small bowl.

Cut a horizontal slit along the thin, long edge of a chicken breast half, nearly through to the opposite side. Open up the breast and place one-fourth of the filling in the center. Close the breast over the filling, pressing the edges firmly together to seal. Repeat with the remaining chicken breasts and filling.

Lightly beat egg white with a fork in a medium bowl. Place breadcrumbs in a shallow glass dish. Hold each chicken breast half together and dip in egg white, then dredge in breadcrumbs. (Discard leftovers.)

Heat oil in a large nonstick skillet over medium-high heat. Add chicken breasts; cook until browned on one side, about 2 minutes. Place the chicken, browned-side up, on the prepared baking sheet. Bake until the chicken is no longer pink in the center or until an instant-read thermometer registers 170 degrees F, about 20 minutes.