Cut the kernels off each cob. Scrape the cobs with the back of a knife to collect the corn pulp. Set aside about ¼ cup of kernels for garnish.
Add corn, pulp, cucumber, pepper, tomatoes, shallot, garlic, jalapeño, olive oil, lemon juice, vinegar, and salt to a blender. Blend until smooth.
If too thick, blend in a splash of water or a little more olive oil.
Refrigerate for at least 2 hours to let the flavors meld.
Top with reserved corn, tomatoes, basil, a drizzle of basil oil, and flaky salt.