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 the butter.
Add the sandwiches to the press and cook over moderate heat until the cheese is melted, about 6 minutes.