Using a juicer, juice the cucumbers into a large bowl (you should have about 3 cups of juice). Stir in the cilantro sprigs, garlic, ginger, 3 tablespoons of the lime juice, 2 teaspoons of salt, and 1/4 teaspoon of pepper. Cover and refrigerate overnight.

Spread 1/3 cup of salt in a medium skillet. Heat the skillet until the salt is hot. Add the shrimp and cook over moderately high heat, turning once, until pink, about 4 minutes. Remove the shrimp and let stand until cool enough to handle, about 5 minutes. Peel the shrimp and remove the intestinal vein. Cut the shrimp into 1/2-inch pieces and transfer to a medium bowl. Add the chopped cilantro, avocado and the remaining 1 teaspoon of lime juice.

Strain the cucumber juice through a fine strainer into a large measuring cup. Spoon the shrimp into shallow bowls and drizzle with the chile oil. Pour the cucumber soup around the chopped shrimp and serve.