Preheat the oven to 375°F.
Put the beets into a small baking dish and rub them with 1 tablespoon of the oil. Cover the dish with aluminum foil and roast for 30 minutes.
While the beets are roasting, peel and cut the carrots into 1-inch-thick rounds, and trim the Brussels sprouts and cut them in half lengthwise. Put the carrots, sprouts, and garlic cloves in a large baking dish and toss with the remaining 2 tablespoons oil. Sprinkle with salt and pepper.
After the beets have been cooking for 30 minutes, add the large pan of vegetables to the oven separately, and cook everything for 1 hour more, stirring the vegetable mixture once or twice.
Remove the beets from the oven and transfer them to a cutting board to cool. Stir the thyme into the carrot and Brussels sprout mixture and let it continue to cook for another 10 minutes while the beets are cooled and cut.
When the beets are cool enough to handle, after about 5 minutes, peel, then cut them into 1-inch chunks. Remove the other vegetables from the oven, toss with the beets, season with salt and pepper, and serve.