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In a small bowl, mix together the yogurt, 1/2 teaspoon of the cumin, 1/2 tablespoon of the lemon juice, 1/2 tablespoon of the cilantro, and the sugar. Set aside the yogurt sauce.
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Preheat the grill or a heavy pan over medium-high heat. In a large bowl, combine the turkey, egg, green onions, curry powder, salt, and the remaining 1 tablespoon lemon juice, 1 tablespoon cilantro, and 2 teaspoons cumin. Mix well, then form into 4 patties.
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Lightly oil the grill or stovetop grill pan and grill the burgers on one side until mostly cooked, 6 to 8 minutes, then flip and grill until cooked through, 3 to 4 minutes longer.
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Place 1 lettuce leaf on the bottom half of each bun. Top each with 2 tablespoons of chutney, some sliced tomato, a burger, a dollop of yogurt sauce, and the bun top and serve.