In a small bowl combine shredded chicken, kimchi, and mayonnaise.
Preheat a small skillet over medium heat. Spread ½ tablespoon butter on one side of a slice of bread. Place 1 slice bread butter-side-down in skillet; place 1 slice provolone and 1 slice Havarti cheese onto bread. Top with chicken mixture, then with remaining slice provolone and remaining Havarti. Butter second bread slice with remaining butter on one side and place butter-side-up on top of sandwich.
Cook sandwich in skillet until bread is lightly browned and flip over, about 2 ½ minutes per side.