In a medium bowl, stir together tuna, cannellini beans, red onion, and garlic. Stir in the mayonnaise. Set aside.
Preheat an electric sandwich press, a covered indoor grill, a grill pan, or a skillet. To assemble sandwiches, spread one side of each bread slice with butter. Spread the plain sides of half of the bread slices with tuna mixture. Top tuna mixture with tomato slices, basil leaves, and cheese. Top with remaining bread slices, buttered sides up.
Place sandwiches (half at a time, if necessary) in the sandwich press or indoor grill; cover and and cook about 6 minutes or until cheese melts and bread is toasted. (If using a grill pan or skillet, place sandwiches on grill pan. Weight sandwiches down and grill about 2 minutes or until bread is toasted. Turn sandwiches over, weight down, and grill until remaining side is toasted.) Makes 4 sandwiches.