To a skillet, add the oil and butter over medium-low heat. When the butter is melted, add the onion, toss to coat in the butter and oil. Toss every 5 minutes until the onions are soft and golden brown. About 30 to 40 minutes.
When the onions are ready. To a bowl, add the beef and season with the Worcestershire sauce. Divide the meat into 4 equal portions and shape into oval patties. Season both sides of the burgers with salt and pepper.
Heat a cast iron skillet or griddle pan over medium heat. Add the burgers to the pan and don’t turn or move until they develop and brown crust, about 3 minutes. Turn and repeat. Remove the burgers to a plate and wipe out the pan. Turn down the heat on medium-low.
Line up 4 slices of rye bread and top with 1 slice of Swiss cheese on each. Add a burger to each, top with caramelized onions then another slice of Swiss. Top with the slices of rye bread and spread an even layer of mayonnaise onto the tops of the rye bread.
Add as many of the sandwiches to the pan that will fit, mayonnaise side down. Spread mayonnaise over the tops of the bread on the sandwiches in the pan. Cover with foil and cook until the cheese starts to melt (make sure the bottom doesn’t burn). Turn and brown the other side.