Heat the oil in a saucepan over medium-high heat.
Add the onion and cook, stirring, until soft, about 5 minutes. Add the cauliflower, broth, and milk and simmer until tender, about 15 minutes.
Cut the rolls in half. Layer the bottoms with the prosciutto and watercress. Drizzle with 1 tablespoon of the extra-virgin oil. Top with the roll halves.
Puree the soup in the pan using a handheld immersion blender or, working in batches, a standard blender. Stir in 1 2/3 cups of the Cheddar, 1 teaspoon salt, and 1/4 teaspoon pepper.
Divide the soup into bowls. Add the remaining extra-virgin oil and Cheddar and 1/4 teaspoon pepper. Serve with the sandwiches.
Tip: The colder and harder Cheddar is, the easier it is to shred. So don't remove the cheese from the refrigerator until you're ready to grate it.