Preheat oven to 400°F. Slice hoagie rolls lengthwise. Brush insides with butter or olive oil. Toast on baking sheet 5 minutes until golden and crisp. Set aside.
On a large cutting board, stack deli meats, cheese, lettuce, tomato, onion, and pepperoncini. Chop everything into small, even pieces (about ¼-inch). Chop in batches if needed.
In a large mixing bowl, whisk Italian dressing, mayonnaise, oregano, garlic powder, and red pepper flakes until smooth.
Add all chopped ingredients to the bowl. Toss with tongs until everything is coated evenly. Add Parmesan and toss again.
Cover and refrigerate 10 minutes for flavors to meld.
Stuff each toasted roll with the grinder salad mixture. Drizzle extra dressing on top. Close rolls and serve right away.