For the burgers: Place a grill pan over medium-high heat or preheat a gas or charcoal grill. Combine the beef, pesto, Parmesan, salt and pepper in a large bowl. Form the mixture into eight 4-inch-diameter patties, each about 3/4-inch thick. Grill for 5 minutes. Turn the burgers over and place a slice of mozzarella on top. Cook for about 5 minutes longer until the burgers are cooked through and the cheese is slightly melted. Allow the burgers to rest for 5 minutes.
For the pesto mayonnaise: In a small bowl, combine the mayonnaise and pesto until smooth.
To assemble the burgers: Drizzle the cut side of the bread with olive oil and grill until lightly toasted, 1 to 2 minutes. Spread the pesto mayonnaise over the bottom halves of the grilled bread. Place the burgers on top and add 2 tomato slices. Add the top halves of the bread and serve.
Yield: 1 1/4 cups.
Bake in a preheated 350 degrees F oven until lightly toasted, 6 to 8 minutes. Cool completely before using. Combine the basil, Parmesan, pine nuts, salt, pepper and garlic in a food processor. Pulse until finely chopped. With the machine running, gradually add the olive oil until the mixture forms a smooth paste.
Freeze the mozzarella for 20 minutes to make slicing easier.
To toast the pine nuts, arrange in a single layer on a baking sheet.