In a medium bowl, mix the tuna with the onion, olive oil, vinegar, basil, and crushed red pepper. Season with salt and pepper.
Heat a panini press or griddle. Spread the cut sides of the rolls with mustard and mayonnaise and top each roll half with a slice of cheese.
Spread the tuna salad on the bottoms and cover with the pickles. Close the sandwiches and spread the outsides of the rolls with butter.
Place the sandwiches on the press or griddle and cook over moderate heat until the cheese is melted, about 6 minutes. To make sure the sandwiches are fully cooked, open them up and place in a 450°F oven for about 1 minute.
Remove from oven, close sandwiches, cut in half, and serve.