Lightly coat a baking sheet with sides with cooking spray.
Lightly beat the egg white with a fork in a medium bowl.
Mix the breadcrumbs and salt in a shallow dish.
Cut a horizontal slit along the thin, long edge of a chicken breast half, nearly through to the opposite side.
Open up each breast and place one-fourth of the cheese filling in the center.
Close the breast over the filling, pressing the edges ¬firmly together to seal.
Repeat with remaining chicken breasts and filling.
In a large nonstick skillet, heat the oil over medium-high heat. Add chicken and cook until browned on one side, about 2 minutes.
Transfer chicken to the prepared baking sheet, browned-side up, and bake until it is no longer pink in the center or until an instant-read thermometer registers 170°F, about 20 minutes.