1. Set a rack in the top third of the oven, and heat the broiler.

2. On a cookie sheet, toast bread halves under the broiler until lightly golden on both sides. Transfer to a work surface.

3. Spread bread with mayonnaise, if using. Top each half with tuna salad and a single layer of cheese.

4. Return the open-faced sandwiches to the cookie sheet, and broil, watching carefully, until the cheese is completely melted, about 1 1/2 minutes. Cut each piece of bread in half for four sandwiches, or into 1- to 2-inch slices for easy nibbling. Garnish with fresh basil, if using, and serve.