How to Make It
In a blender, combine the cucumber, avocados, buttermilk, vinegar, sugar, 1/2 tablespoon of the lime juice and a pinch of salt. Blend until smooth. Strain the soup through a fine sieve and refrigerate until chilled, about 40 minutes.
In a large saucepan, heat 2 tablespoons of the olive oil. Add the shrimp shells, garlic, shallot and carrot and cook over moderate heat, stirring until the shells are lightly browned, and the vegetables are softened, about 8 minutes. Add the peppercorns, coriander seeds, crushed red pepper and cumin seeds and cook, stirring, for 1 minute. Add the tomato paste and cook, stirring, for 1 minute. Add the water, honey and remaining 1 tablespoon of lime juice and bring to a boil. Simmer over moderate heat until reduced to 1/2 cup, about 15 minutes. Strain the shrimp sauce into a heatproof bowl.
In a medium skillet, heat the remaining 1 tablespoon of olive oil until shimmering. Add the shrimp, season with salt and pepper and cook over high heat for 1 minute. Turn the shrimp, add the shrimp sauce and simmer just until the shrimp are cooked through, about 2 minutes. Swirl in the butter.
Place the shrimp in soup bowls, spoon the cucumber soup all around and drizzle the shrimp sauce on top. Sprinkle with the mint and serve right away.
If you are short on time, the cucumber-avocado soup is delicious on its own. Just follow the recipe through step 1, then serve with a drizzle of olive oil and chopped mint.