Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add the onion and cook, stirring occasionally, until slightly softened, about 3 minutes. Add the garlic and cook, stirring occasionally, until soft, about 3 more minutes. Season with salt and pepper, transfer to a large bowl and set aside.
Lightly pound the steak between 2 sheets of plastic wrap until about 1/8 inch thick. Heat the remaining 2 tablespoons olive oil in the same skillet over medium-high heat. Add the steak in batches and cook until browned, about 30 seconds per side; season with salt and pepper. Transfer the steak to the bowl with the onion and garlic. Add the parsley and giardiniera brine and toss to coat. Drizzle the cut sides of the rolls with olive oil. Pile the steak and onion mixture on the roll bottoms and top with the giardiniera and provolone. Cover with the roll tops. Serve with hot sauce, if desired.