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Combine radish, carrot, vinegar, and sugar in a medium bowl; mix well. Marinate at room temperature, covered, for 15 minutes. Drain well before using.
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Combine yogurt and 1 Tbsp. pepper sauce in a small bowl; mix well. Set aside.
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Preheat grill or broiler to high.
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Combine turkey, remaining 2 tsp. pepper sauce, cornstarch, fish sauce, garlic, basil, and oil in a medium bowl. Season with salt and pepper if desired; mix well with clean hands. Form turkey mixture evenly into four patties.
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Grill or broil patties for about 5 minutes on each side, or until no longer pink in the middle.
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Place each patty on a lettuce leaf. Top evenly with yogurt sauce, radish mixture, jalapeno (if desired), and cilantro.