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In a small bowl, combine yogurt, curry powder, 1/4 teaspoon salt, and lime juice. Let sit 15 minutes.
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In a food processor, pulse chickpeas, scallions, and 1/2 teaspoon salt until a chunky paste forms. In a large bowl, stir together chickpea mixture, rice, carrot, breadcrumbs, cilantro, and 2 tablespoons yogurt mixture. Divide into 4 patties. (To freeze, wrap each patty in plastic and freeze, up to 2 weeks. Let thaw before cooking.)
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Heat oil in a nonstick skillet over medium. Cook patties until lightly golden and heated through, about 4 minutes per side. Serve topped with remaining yogurt sauce and cilantro leaves.