Preheat the grill to medium-high heat.
Season the elk tenderloins with salt, pepper, and a drizzle of olive oil. Rub to coat evenly.
Grill the tenderloin: Place the elk on the grill and cook for 4-5 minutes per side for medium-rare. Adjust time for desired doneness. Use a meat thermometer to check for an internal temperature of 130°F for medium-rare. Prepare the balsamic glaze: In a small saucepan, combine balsamic vinegar and honey. Bring to a simmer and cook until the mixture has reduced by half, becoming syrupy (about 10 minutes). Baste the elk: During the last minute of grilling, brush the elk tenderloins with the balsamic glaze.
Rest the meat: Remove the tenderloins from the grill and let them rest for 5-10 minutes to lock in juices. Serve: Slice the elk tenderloin and drizzle with any remaining balsamic glaze. Garnish with fresh rosemary or thyme, if desired.