Preheat the oven to 350° F. Place the whole beets in a roasting pan with the thyme sprigs, garlic cloves, cardamom, and peppercorns. Drizzle 2 tablespoons of olive oil over the beets and add 1/3 cup of water to the pan. Cover with tin foil and roast in the oven until tender and the skins slip easily off the beets, approximately 1 hour.
While the beets are cooking, cut the turnips into quarters and slice the carrots into 1- to 1 1/2-inch pieces. Toss the vegetables with 3 tablespoons of the remaining olive oil and salt and pepper.
When the beets are done, remove them from the oven, and turn up the heat: You want a temp in the 450° to 500° F range. Place the carrots and turnips in the hot oven on a roasting pan (go ahead and use the beet pan if you don’t want to wash too many pans). Meanwhile, peel the beets and cut into quarters.
Whisk the rest of the olive oil with the lime juice and harissa. When the carrots and turnips are roasted and browned (about 20 to 30 minutes), toss with the beets. Dress with the harissa-lime juice mixture, toss with the cilantro, and taste and adjust for salt. Serve immediately or let cool to room temperature.