1. Sprinkle chicken evenly with 1 tsp. 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.
2. Slice hard-cooked eggs in half lengthwise; carefully remove yolks, keeping egg white halves intact. Reserve yolks for another use.
3. Stir together mayonnaise, next 4 ingredients, and remaining 1/2 tsp. salt in a large bowl.
4. Pulse cooled chicken, in batches, in a food processor 3 to 4 times or until shredded; stir into mayonnaise mixture until blended. Spoon chicken mixture evenly into egg white halves. Cover and chill at least 1 hour.