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Preheat the oven to 450°F. Soak the quartered figs in a bowl of warm water for 15 minutes, then drain and reserve.
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Trim and discard the chard ends. Remove the chard ribs and slice them into 1-inch pieces. Coarsely chop the leaves, reserving the ribs and leaves separately.
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Place the chicken, skin side up, on a rimmed baking sheet, drizzle with 1 tablespoon of the olive oil, and season all over with 1 teaspoon salt and 1/4 teaspoon pepper.
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In a small bowl, combine the cumin, pumpkin pie spice, thyme, and cayenne and sprinkle all over the chicken.
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Add the shallots, sweet potato, chopped chard ribs, and garlic to the baking sheet. Drizzle with the remaining 1 tablespoon olive oil, season with salt and pepper, and gently toss to coat. Roast for 10 minutes, lower the heat to 400°F, and continue roasting until the chicken is cooked through and the vegetables are tender, turning the vegetables once or twice and adding the figs halfway through cooking, about 22 minutes total.
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Transfer the chicken and vegetables to a plate. Keeping 3 tablespoons pan juices from the baking sheet (spoon off and discard any excess), place the chard leaves on the baking sheet, lightly toss with the pan juices, and roast until softened, about 3 minutes. Season with salt and pepper. Transfer the chard and the other vegetables to a serving platter, top with the chicken, and serve.