Preheat the oven to 400°. In a small bowl, combine the 1/4 cup of olive oil with the minced garlic, rosemary and thyme.
Score the fatty top side of the lamb about 1/4 inch deep. Season the lamb all over with salt and pepper. On a large rimmed baking sheet, toss the whole garlic cloves, onion, carrots and celery with a generous drizzle of olive oil. Arrange the vegetables in an even layer and season with salt and pepper. Scatter the rosemary sprigs over the vegetables and set the leg of lamb on top, fatty side up. Spread half of the garlic-and-herb rub all over the lamb, making sure to rub it into the score marks. Roast the leg of lamb for 20 minutes.
Spread the remaining rub over the lamb and add 1/4 cup of water to the baking sheet. Roast the lamb for about 1 hour and 20 minutes longer, rotating the baking sheet a few times, until an instant-read thermometer inserted in the thickest part of the meat registers 145° for medium. Add a few more tablespoons of water to the baking sheet at any point if the vegetables start to get quite dark.
Transfer the lamb to a carving board and let it rest for 20 minutes. Discard the rosemary sprigs. Carve the leg of lamb into 1/2-inch-thick slices and serve with the roasted vegetables.