Working with half of the ingredients at a time, blend gazpacho in 2 batches in a food processor or blender until fairly smooth. Combine in one container and chill until ready to serve.
Heat a griddle over medium heat. Butter bread slices on 1 side then build sandwiches with buttered sides facing out: cheese, sliced jalapeño and chopped tomato, more cheese and bread. Cook sandwiches until golden and melty, sprinkle with smoked salt and cut on a diagonal. Serve chilled mugs of soup alongside.