In a small bowl, combine the chile, shallot (or onion), garlic, vinegar, lemon juice, mustard, parsley, basil, thyme and sugar. Whisk in the EVOO in a slow stream to thicken; season with salt and pepper. Transfer the dressing to a squirt bottle or to a measuring cup for pouring.
Pile the cheese, roast beef, salami and soppressato onto the rolls. Top with the giardiniera and fixins. Dress liberally with the sauce.