BURGERS: Preheat an outdoor grill to medium-high heat.
In a large bowl, combine ground beef, pesto, Parmesan, Montreal steak seasoning. Form the mixture into four 4-inch-diameter patties, each about 3/4-inch thick.
Grill, covered 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.
PESTO MAYONNAISE: In a small bowl, stir together the mayonnaise and pesto until smooth.
ASSEMBLE THE BURGERS: Grill the cut side of the buns until lightly toasted, about 1 to 2 minutes. Spread the pesto mayonnaise over the bottom of the grilled buns. Top with burgers, tomato slices, and some basil leaves. Cover with top buns and serve.
HOMEMADE BASIL PESTO: Combine the basil in with the pine nuts, pulse a few times in a food processor. (If you are using walnuts instead of pine nuts and they are not already chopped, pulse them a few times first, before adding the basil.) Add the garlic, pulse a few times more.
Slowly add the olive oil in a constant stream while the food processor is on. Stop to scrape down the sides of the food processor with a rubber spatula. Add the grated cheese and pulse again until blended. Add a pinch of salt and freshly ground black pepper to taste. Makes about 1 1/4 cups.