Drain the tuna as completely as you can. Flake into a bowl and top with the Greek yogurt, pickles, red onion, lemon juice, mustard, parsley, salt, and pepper. With a fork, stir to combine. Taste and adjust the seasoning as desired.
For each melt, top a slice of bread with one-third of the tuna mixture, spreading it evenly all the way to the edges. Lay a slice of cheese on top, then another slice of bread. Butter the top of the bread with 1/2 tablespoon butter. (If cooking in a panini press or the air fryer, go ahead and butter the other side of the melt too.)
To Cook on the Stove: Heat a nonstick skillet over medium low. Once the skillet is hot, add the sandwich, buttered-side down. Butter the top. Cook on each side for 2 to 4 minutes until the bread is toasty and golden—low and slow will guarantee the most melty, perfectly cooked grilled cheese.
To Cook in a Panini Press: Cook on medium-high until the bread is golden and the cheese melted.
To Cook in the Air Fryer: Use toothpicks to secure the tuna melt by sticking a toothpick into two of the opposite corners of the bread. Air fry at 360°F for 9 to 12 minutes, flipping halfway through.
Let the tuna melt cool for a few minutes. Cut in half and enjoy.
