Combine mayonnaise and pesto, stirring well. Spread 1 tablespoon mayonnaise mixture on each of 4 bread slices; top each slice with 2 ounces turkey, 1/2 ounce cheese, and 2 tomato slices. Top with remaining bread slices.
Preheat grill pan or large nonstick skillet coated with cooking spray over medium heat. Add sandwiches to pan; top with another heavy skillet. Cook 3 minutes on each side or until golden brown.
In Italian, "panini" means small bread and refers to a pressed sandwich. Using a grill skillet gives the sandwich appetizing grill marks, but the recipe works just as well in a regular nonstick skillet. If you don't have provolone cheese, you can use mozzarella.