Assemble the wrap: Lay the lavash, flatbread, or tortilla on a clean work surface. Arrange the provolone cheese slices down the center, followed by the romaine lettuce leaves.
Add the vegetables: Spread the giardiniera, olives, sun-dried tomatoes, and roasted red peppers evenly over the top. Add the sliced red onion and tomato. Sprinkle on the dried oregano and drizzle the olive oil and balsamic vinegar over the top.
Roll the wrap: Fold one side of the wrap inward, then roll it tightly.
Serve: Slice the wrap in half and serve immediately. For a crispier exterior, lightly toast the assembled wrap in a dry skillet or panini press for 2–3 minutes per side before slicing.