Heat gas or charcoal grill. Between pieces of plastic wrap or waxed paper, place each chicken breast smooth side down; gently pound with flat side of meat mallet or rolling pin until about 1/4 inch thick.
In small bowl, mix mayonnaise and mustard. Cut 4 (18x12-inch) sheets of heavy-duty foil. Place chicken breast on center of each sheet. Spread each with about 1 tablespoon mayonnaise mixture and top each with 1 slice ham and 1 slice cheese. Bring up foil sides so edges meet. Seal edges, making tight 1/2-inch fold; fold again, allowing space on sides for heat circulation and expansion.
Place packets, seam side up, on grill. Cover grill; cook over medium heat 20 to 30 minutes, rearranging packets several times during cooking, until juice of chicken is clear when center of thickest part is cut (170°F).
To serve, cut large X across top of each packet; carefully fold back foil to allow steam to escape.