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Adjust rack to top position (top and middle positions for 4 servings) and preheat oven to 425 degrees. Wash and dry all produce.
Halve bell pepper; remove stem and seeds. Cut potatoes into ½-inch-thick wedges. Halve, peel, and finely chop shallot. Peel and mince or grate ginger. Trim and thinly slice cucumber into rounds. Halve buns.