Heat 1 tablespoon of 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 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 ⅔ cups of the Cheddar, 1 teaspoon salt, and ¼ teaspoon pepper.
Divide the soup into bowls. Add the remaining oil and Cheddar and ¼ teaspoon pepper. Serve with the sandwiches.