Prep the vegetables: Chop the romaine, cucumber, tomatoes, red onion, and roasted red peppers into small, bite-friendly pieces and place them in a large mixing bowl. Pat the wetter ingredients dry with a paper towel if needed.
Build the creamy Italian dressing: In a smaller bowl, stir together the mayonnaise, Greek yogurt or sour cream, red wine vinegar, olive oil, Italian seasoning, garlic, crushed red pepper flakes, black pepper, and a small pinch of salt. Mix until smooth and fully blended. Taste and adjust seasoning if needed.
Combine and chill: Pour the dressing over the vegetables in the large bowl. Add the olives, mozzarella or provolone, and Parmesan. Gently mix everything until evenly coated. Cover and refrigerate the filling for at least 15 minutes to let flavors meld.
Toast the bread: If using, lightly butter or oil the bread and toast it in a skillet or oven until golden and crisp. This step is optional but recommended for better texture.
Assemble the sandwiches: Spoon the chilled filling onto the bottom halves of the rolls or bread slices. Add optional salami or turkey if desired. Top with the other halves of the bread. Serve immediately.