Step 1: With a sharp paring knife, make holes at 1-inch intervals all over the roast. Stuff 1/3 of the holes with the asiago, 1/3 with the garlic slivers, and 1/3 with the rosemary sprigs.
Step 2: Generously season the roast with salt and pepper.
Step 3: Set up the rotisserie following the manufacturer’s instructions. If using a gas grill, preheat the front and rear burners to high. If using a charcoal grill, light the coals and rake into rows in front and back, leaving a gap in the center. Skewer the roast on the spit. Rotisserie the roast until cooked to medium-rare (about 145°F on an instant-read thermometer; 1-1/4 to 1-1/2 hours on a covered rotisserie. For medium (160°F), cook for 1-1/2 to 2 hours. If using a charcoal grill, replenish the coals as needed.
Step 4: Transfer the roast to a cutting board and let rest for 10 minutes. Cut into thin crosswise slices using an electric knife or sharp carving knife. Serve immediately.