In a large bowl, combine the minced meat, steak sauce, garlic, salt, and pepper and stir until smooth.
Turn on the grill to preheat to medium heat. Form 8 small cutlets from the resulting mass. In a small bowl, combine mozzarella and pesto. Flatten each cutlet and put about 1/4 cup of cheese mass in the middle of 4 cutlets, cover the filling with the remaining cutlets and fasten, forming whole cutlets with filling inside.
Place the cutlets on the grill rack, cover the grill with a lid and cook for about 5 minutes on each side (if you check the temperature inside the cutlets with a meat thermo, it should reach 165 degrees.)
Put ready cutlets between slices of rye bread together with lettuce, fasten with skewers.