Heat 2 tablespoons vegetable oil in heavy large pot over medium heat. Add half of onions and sauté until translucent, about 5 minutes. Add corn cobs and 8 cups water. Simmer until liquid is reduced to 4 cups, about 25 minutes. Cool slightly. Refrigerate overnight. Strain corn broth; discard cobs.
Heat remaining 2 tablespoons oil in another heavy large pot over medium heat. Add remaining onions and sauté until soft, about 5 minutes. Add corn kernels and sauté until almost tender, about 15 minutes. Add strained corn broth and remaining 2 cups water. Simmer uncovered 40 minutes. Cover and simmer until corn is very tender, about 20 minutes longer. Cool slightly.
Working in batches, puree soup in blender until smooth. Return soup to pot. Stir in cream. Season to taste with salt and pepper. (Can be made 1 day ahead. Cool slightly, then cover and refrigerate. Bring to simmer before continuing.) Divide soup among 6 bowls. Sprinkle with basil and serve.