Place giardiniera in a food processor and pulse into finely chopped relish. Transfer to a bowl and reserve. Wipe out processor and return to base.
For the dressing, add vinegar, shallot or onion, garlic, salt, pepper, sugar, chili pepper and herbs to the bowl. Turn on processor and stream in EVOO. Process until smooth.
Line sub rolls with cheese, fill with meats and top with lettuce, tomato and onion. Pour dressing over sandwiches and top with spicy vegetable relish. Set sub top in place and serve.