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Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Dice potatoes; place in a medium pot with enough salted water to cover by 2 inches. Trim, peel, and cut carrots into 1-inch-thick pieces on the diagonal. Trim and thinly slice scallions, separating whites from greens. Halve, peel, and mince shallot until you have 3 TBSP. Halve lime.