How to Make It
Using a thin knife and working in a bowl, cut the kernels from the cobs; you should have 6 cups. Working over another bowl, scrape the cobs with the back of a knife to release the corn milk; discard the cobs.
In a large skillet, heat 3 tablespoons of the olive oil over moderately high heat. Add the onion and garlic and cook, stirring occasionally, until softened, about 4 minutes. Add 4 1/2 cups of the corn kernels and the corn milk and cook, stirring occasionally, until the corn is crisp-tender, about 6 minutes. Season with salt and pepper. Transfer to a blender and let cool slightly. Wipe out the skillet.
Add 1 1/2 cups of water to the blender with the corn and puree at high speed until very smooth, about 5 minutes. Strain the puree through a fine sieve set over a large bowl, pressing on the solids. Discard the solids. Return the corn puree to the blender and add the spinach, chopped jalapeños and 1 tablespoon of the lime juice and puree until the spinach is finely chopped and the soup is green. Transfer to a large bowl and stir in 1 1/2 cups of water. Cover and refrigerate the soup until cold, at least 2 hours.
Meanwhile, in the same skillet, heat the remaining 1 tablespoon of olive oil. Add the remaining 1 1/2 cups of corn kernels and cook over moderately high heat, stirring occasionally, until crisp and golden, 1 minute. Transfer the corn to a small bowl and refrigerate until cold.
Add the sliced jalapeños and the remaining 1 teaspoon of lime juice to the chilled corn kernels and season with salt and pepper. Transfer the soup to bowls and top with the corn. Drizzle with olive oil and serve.