How to Make It
Heat oven to 400° F. Arrange the salami in a single layer on a rimmed baking sheet and bake until crisp, 8 to 10 minutes; let cool.
Heat grill to medium-high. Form the beef into four ¾-inch-thick patties. Use your fingers to make a shallow well in the top of each patty. (This will prevent overplumping during cooking.) Season with a total of ½ teaspoon each salt and pepper.
Oil the grill and cook the patties with the wells facing up for 4 to 5 minutes. Flip and cook until an instant-read thermometer inserted in the center registers 155° F for medium, 4 to 5 minutes more. Top the burgers with the mozzarella during the last 3 minutes of cooking. Dividing evenly, spread the rolls with the tomato pesto. Form sandwiches with the rolls, burgers, salami, and arugula.