Place the avocados and buttermilk in a food processor or blender and process until smooth. Transfer to a large, non-reactive bowl. Stir in the ice water, lime zest, 1 1/2 teaspoons salt, and cumin. Taste and season with salt as needed since chilled soups often need extra seasoning. The soup is best made and served immediately so that the bright green color does not start to turn gray.
In a medium skillet over medium heat, heat the oil until hot.
Add the corn and onion and cook, stirring often, 4 minutes. Stir in the cumin and 1/4 teaspoon salt and cook until the onions have softened slightly and the corn is still crisp, 1 minute more. Stir in the lime juice. Transfer to a non-reactive bowl and cool to room temperature. (Relish can be prepared one day ahead to this stage; cover and refrigerate. Bring to room temperature before using.) Stir in 2 teaspoons of the cilantro. Taste and add more salt if needed.
Ladle the soup into bowls. Garnish each serving with some corn relish and a sprinkle of cilantro. Pass extra relish in a small bowl.