For the tuna: In a large bowl, break up the tuna with a fork and add the mayonnaise, lemon juice, red onions, green onions, basil, cilantro, parsley, mint and vinegar. Mix well to combine. Taste and season with salt, pepper and extra lemon juice and mayonnaise as desired.

For the sandwich: Brush the softened butter on 1 side of each slice of bread, setting aside some butter for cooking the sandwiches. Place the bread buttered-side down on a baking sheet. Divide the tuna among 4 slices of bread and top each with 2 slices of cheese. Complete the sandwich with the remaining 4 slices of bread, buttered-side up.

Heat a large nonstick skillet over medium-high heat and add half of the remaining butter. When the butter melts, place 2 sandwiches in the pan and cook until toasted and golden brown on each side. Repeat with the remaining sandwiches. (If your pan is large enough, you can cook all the sandwiches at the same time.) Serve warm, with pickles.