With the rack in the middle position, preheat the oven to 400°F (200°C).
In a small saucepan over medium-high heat, brown the onion and garlic in the oil. Add the spices and continue cooking for 30 seconds. Off the heat, stir in the remaining ingredients. Set aside.
In a 13 x 9-inch (33 x 23 cm) baking dish, place the chicken legs. Drizzle with the sauce. In a second baking dish of about the same size, toss the Brussels sprouts, onions and carrots with the oil. Season with salt and pepper.
Bake both dishes side by side. Cook the chicken for about 1 hour or until the meat pulls easily from the bone, and cook the vegetables for about 45 minutes or until tender and thoroughly roasted, stirring halfway through cooking.
Serve the chicken with the vegetables and drizzle with the cooking juices.