Bring a few inches of water in a deep skillet to a boil. Season with salt and add the rapini/broccoli rabe; cook 3 minutes and drain. Return the pan to the heat and add the olive oil, three turns of the pan. Add the red pepper and garlic and stir for a minute. Add the broccoli rabe and toss for 2-3 minutes more; adjust the salt, to taste.
Heat a cast iron skillet or griddle pan over medium-high to high heat.
Combine the sausage and beef with the Worcestershire, grated onion, salt and pepper. Form four patties, making them thinner at the center and thicker at the edges for even cooking. Drizzle the burgers with olive oil and cook the patties for 10 minutes, turning occasionally. Top with the cheese in the last minute or two and melt under a foil tent.
Place the giardiniera relish on the focaccia or roll bottoms. Top with a burger patty, some broccoli rabe and potato chips; set the focaccia/roll tops in place.