Melt 1 tablespoon butter in a nonstick or cast-iron skillet over medium heat. Add kimchi. Cook, stirring constantly, until kimchi starts to dry out, about 1 minute. Whisk eggs, mayonnaise, and salt together in a medium bowl. Add egg mixture to pan and gently scramble with the kimchi, taking care not to break the eggs up too much. When eggs are cooked through, transfer to a plate.

Wipe out pan and return to medium heat. Melt half the remaining butter and swirl to cover the pan. Add 1 tortilla and coat with butter. Spread a quarter of the scrambled eggs on 1 half of the tortilla and fold the other half over. Add the second tortilla to the pan, coat with butter, fill with eggs, and fold over.

One at a time, open the quesadillas, sprinkle a quarter of the grated cheese over the eggs, then close the quesadillas. After 2 minutes, flip quesadillas and cook for a few minutes more before transferring to plates. Repeat with remaining tortillas and ingredients. Cut each quesadilla in half and serve hot.