Heat gas grill to medium or charcoal grill until coals are ash white. Place coals to 1 side in charcoal grill.
Combine giardiniera and Italian dressing in small bowl. Set aside.
Open buns; place cut-sides down onto grill. Grill 1-2 minutes or until toasted. Turn buns over; immediately add 2 slices cheese to each bun. Place opposite heat on grill; keep warm.
Place hot dogs onto grill. Grill, turning occasionally, 5-9 minutes or until heated through.
Place grilled hot dogs into buns; top each with 1/4 cup giardiniera mixture. Serve immediately.
Giardiniera is a combination of assorted pickled vegetables and can be found in jars in the condiments section of the supermarket.