Roughly chop the cucumbers and place them in a large mixing bowl. Add the salt, toss, and set aside while you prepare the other vegetables, or for at least 10 minutes. Remove the husks from the tomatillos and rinse them. Trim the stems, then roughly chop the tomatillos and add them to the bowl with the cucumbers. Toss, then transfer to a blender. Add 1¾ cups of the corn kernels, reserving the rest for serving, along with the jalapeño, onion, garlic, and lime juice.
Blend on high, until smooth, 30 seconds to 1 minute. (If using a powerful blender such as Vitamix, blend on medium to maintain some texture. We want this to be a soup, not a juice!)
Add the basil, olive oil and black pepper to taste and blend on high for 10 seconds. Pour the soup through a mesh strainer, discarding the solids, and taste for seasonings.
Chill the soup at least one hour, or overnight, and serve cold, garnished with a drizzle of oil, the reserved corn kernels, and torn fresh basil leaves.