Heat the oven to 425°F. Slice off the tip of the garlic head to expose the cloves. Rub with a little olive oil and season with salt and pepper. Place on two squares of heavy-duty aluminum foil and sprinkle with a bit of water. Pinch the edges of the foil together to make a packet and place in the oven while its preheating. The garlic will continue to roast along with the lamb and vegetables, and will need to be checked periodically so it doesn’t burn.
Combine the onion, yellow squash, green beans, and cherry tomatoes in a large bowl. Season with salt and pepper and toss with 2 tablespoons of olive oil. Spread the vegetables on a large, lightly oiled rimmed baking sheet. Transfer the garlic packet to the vegetable pan. Roast in the oven for 15 minutes; turn the vegetables once.
Meanwhile, use a boning knife to remove any scraps of meat or fat still left on the rib bones. To make the rack look its best, remove the outer layer of fat, the small bit of meat underneath that layer, and the fat underneath the meat you just removed. Slide the boning knife between the flesh and silver skin to remove that membrane.
In a heavy skillet, heat the vegetable oil over high heat. Season the rack of lamb with salt and pepper. Add the rack meat-side down with ribs facing outward. Sear until well-browned, about 4 minutes. Stand the rack up in the skillet to brown the bottom side, about 2 minutes more.
Use tongs to transfer the lamb to a rimmed baking sheet. In a small bowl, combine the minced garlic, rosemary, and remaining 1 tablespoon olive oil and spoon this over the lamb rack. Remove the pan of vegetables from the oven (whether or not 15 minutes has elapsed) and place the lamb in the oven. If the garlic head is not yet soft, return it to the oven with the lamb. Roast the lamb for 12 to 15 minutes, turning once, or until the meat is medium-rare and an instant read thermometer inserted into the centre of the rack registers around 135°F. Remove from the oven, tent with aluminum foil, and let rest for 5 to 10 minutes.
While the lamb rests, reduce the oven temperature to 375°F and return the vegetables to the oven to finish cooking (check the garlic for doneness). Continue to roast the vegetables until they are softened and nicely caramelized, up to 10 minutes longer.