Make the tuna salad by combining all tuna salad ingredients.
Assemble the sandwiches by spreading a thin layer of mayo on one slice of each sandwich.
Top with 1/4 of the tuna, avocado, and cheese, then remaining slice of bread.
Spread additional mayo or butter on the outside of the sandwich.
Preheat panini maker or a skillet over medium heat.
Place sandwich on panini maker or skillet and cook until golden brown on each side and cheese is melted, about 8-10 minutes. If making it on a skillet, flip after about 4 minutes.
Serve and enjoy!