In a medium bowl, combine the drained tuna with finely chopped celery, red onion, and fresh parsley. Add mayonnaise, Dijon mustard, and lemon juice, seasoning with salt and pepper to taste. Mix well until all ingredients are fully integrated.

For broiled version, preheat your broiler to high. If baking, set the oven to 400°F. Toast each side of the bread slices on a baking sheet until golden brown, about 2-3 minutes per side, or spread butter on one side of each slice and bake for 5-7 minutes butter-side up.

Spread a generous layer of the tuna salad mixture over each slice of toasted bread. Top each open-faced tuna melt with slices of cheese.

If broiling, place the assembled tuna melts under the broiler for 3-5 minutes until the cheese melts and bubbles. For the baking method, place them in the oven and bake for 8-10 minutes.

Remove the tuna melts from the oven or broiler and let cool briefly. Optionally sprinkle with additional seasoning before serving.