Heat the oven to 425°F and position a rack near the top. Cut the carrots diagonally into 1-inch pieces. Put them in a 9 by 13–inch baking pan and scatter the onion and olives on top.
Combine the cumin, cinnamon, and turmeric in a small bowl with a generous sprinkle of salt and pepper; toss to combine. Sprinkle half the spice mixture over the vegetables. Put the chicken on top of the vegetables and rub all over with the remaining spices; turn them skin side up. Tuck the lemons here and there. Pour 3 cups water into the pan.
Transfer the pan to the oven and bake until the chicken skin is browned and crisp and the meat is no longer pink at the bone, 30 to 40 minutes. Garnish with the cilantro, and serve hot directly from the pan with a big spoon to get all the juices.