In a food processor, chop giardiniera into a fine relish; transfer to a bowl.
Freeze the meat for 5 minutes to make it easier to slice.
Heat a tablespoon of oil in large, nonstick skillet over medium-high heat, a turn of the pan. Add the peppers, onions and garlic, season with salt and pepper, cook to tender, 10-12 minutes. Add stock to keep them moist.
For sauce, melt the butter over moderate heat in a saucepot. Add flour, granulated garlic and onion, and whisk in milk. Thicken sauce, season with salt and white pepper and stir in the provolone to melt. Turn heat to warm/low.
Heat large, cast-iron pan or griddle over medium-high to high heat. Very thinly slice the steak on a bias against grain. Drizzle meat with olive oil and flash grill it in small batches to crisp at edges and cooked through.
Assemble hoagies: In the toasted rolls, place giardiniera relish, meat, provolone sauce, peppers and onions.