Sprinkle chicken evenly with salt and 1/2 tsp. pepper. Grill, covered with grill lid, over high heat (400 to 500 ) 6 to 8 minutes on each side or until done. Let stand 15 minutes; cover and chill at least 30 minutes.
Slice hard-cooked eggs in half lengthwise; carefully remove yolks, keeping egg white halves intact.
Stir together all ingredients, and remaining salt in a large bowl.
Pulse cooled chicken, in batches, in a food processor 3 to 4 times or until shredded; Spoon chicken mixture evenly into egg white halves. Cover and chill at least 1 hour.