Heat a large soup pot, add the oil, and then sauté the onions and carrots until softened, about 8-10 minutes.
Add the broth and water and bring to a boil.
Add the cauliflower, potatoes, half of the basil leaves, and 2 teaspoons salt and cook until vegetables are soft.
Puree the remaining 1/4 basil with 1/2 cup of the cooking liquid in a blender. Add 3 cups of the soup to the blender and puree.
Return the puree to the pot and season with salt and pepper to taste. Add the corn and cook an additional 3 minutes.