Stack deli meats, cheese, and veggies on a large cutting board. Chop everything into small, bite-sized pieces (about ¼-inch). Work in batches if needed for even chopping.
In a large mixing bowl, whisk together Italian dressing, mayonnaise, oregano, and red pepper flakes until smooth.
Add all chopped ingredients to the bowl with the dressing. Toss with tongs or gloved hands until everything is evenly coated. Taste and add more dressing if needed.
Cover bowl and refrigerate 10 minutes to let flavors meld.
Slice hoagie rolls lengthwise. Stuff each roll with generous amount of chopped mixture. Drizzle extra dressing on top. Close rolls and serve immediately.