Once the lamb is cooked to desired doneness, it will stay warm on a cutting board under a tent of foil until you begin slicing.
Remove any string or netting and lay lamb out flat on a cutting board. Trim any excess fat, taking care not to cut any large holes in the meat. With lamb arranged bone side up and fat side down, spread the surface with tapenade and roll the meat into a cylinder. Some tapenade may spill out of the roll. Using butcher twine, secure the rolled lamb by tying loops of twine at 1-½-inch intervals along its length. Collect any tapenade that may have squeezed out and rub on top. Using the point of a sharp paring knife, make incisions all over the lamb and stuff each one with a garlic sliver and small rosemary sprig. The lamb may be prepared several hours ahead up to this point. Refrigerate if you plan to wait more than 1 hour before roasting.
Preheat the oven to 350 degrees. If the lamb has been refrigerated, let it sit at room temperature about 30 minutes. Coat a roasting pan with cooking spray. Place the lamb in the pan, then arrange shallots and carrots around it. Season with salt and pepper, then drizzle with olive oil. Scatter any leftover rosemary over the vegetables, then toss to coat. Roast in the lower third of the oven, stirring the vegetables once or twice, until the meat reaches 145 degrees on an instant-read thermometer (for medium rare) or 160i¿½ (for medium), about 1-½ to 2 hours. Cover loosely with foil and let rest 20 minutes. Remove the strings and carve into ½-inch-thick slices for serving.