Whisk olive oil, red wine vinegar, oregano and black pepper in a large bowl. Add shaved fennel, dry radicchio and drained cherry peppers and toss just before filling the rolls.
Split the rolls lengthwise, leaving a hinge if possible. Pull out a small amount of loose center crumb only if the rolls are very dense.
Divide provolone among the bottom halves, then fold mortadella, soppressata and bresaola evenly along each roll. Keep the meat in shallow folds rather than a high central pile.
Lift the dressed vegetables from the bowl with tongs, leaving pooled dressing behind, and divide over the meat. Close each roll gently.
Wrap snugly in parchment and rest for 10 minutes, then cut across the middle. Serve one sub per person or refrigerate promptly for later the same day.