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 Rocket Patties with the wells facing up for 4 to 5 minutes. Flip and cook 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.