Spread both slices of bread with the butter. Spread the mayo and pesto on the opposite side of the butter. Top with chicken, mozzarella, and tomato. Sprinkle with salt. Top the sandwich with the other piece of buttered bread.
Place on the panini grill and close the top. Grill until the cheese begins to get melty. Turn the grill up to medium to get a nice crunch on the outside of the bread.
Remove from the grill, slice in half, and serve hot.