You’ll need 12 wooden toothpicks coated with nonstick spray. In a small bowl, mix apple, cheese, bread crumbs, chopped thyme, lemon juice and half the salt mixture. On a cutting board, lightly press each chicken breast flat with one hand; using a sharp knife, carefully cut into side of breast to form a deep horizontal pocket (do not pierce top, bottom or far side of breast).
Divide the apple mixture into 4 portions and stuff into chicken pockets. With the prepared toothpicks, pin closed. Sprinkle the remaining salt mixture over breasts. Dredge chicken in flour to lightly coat; tap off excess flour. (At this point, chicken can be refrigerated, covered, up to 6 hours.)
Heat oil in a large nonstick skillet over medium-high heat. Add chicken and brown 3 minutes per side. Pour 1⁄2 cup of the apple cider and the broth into skillet; add thyme sprigs. Bring mixture to a boil; reduce heat to low, cover and simmer 5 to 7 minutes until chicken is cooked through and stuffing registers 165ºF on an instant-read thermometer.
Remove chicken to a serving plate; cover with foil to keep warm. Whisk mustard and remaining 1⁄4 cup cider into juices in skillet; boil mixture 3 minutes on high until reduced and slightly thickened. Spoon sauce over chicken and serve.