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In a large bowl, mix together lobster meat, jicama, scallions, and Lychee-Mango Pu-ree. Season with salt and pepper. On salad plates, lay out 3 leaves of butter lettuce and top with a small mound of lobster salad. Drizzle with extra puree. Garnish with the remaining scallions.
In a saute pan over high heat, saute onion, ginger, and jalapeno peppers in 2 tablespoons oil until soft, about 4 minutes. Season with salt and pepper. Add mangoes and lychees and cook 3 minutes, until hot. Transfer to a blender and puree at a high speed in batches. Add lime juice and slowly drizzle in the remaining ⅓ cup oil. Transfer to a glass jar, cover, and store in the refrigerator up to 2 weeks.