Warm a large cast iron skillet over medium-high heat, then add olive oil and butter. Once melted, add onion slices and 1 tsp. salt. Stir the onions and spread out evenly over the pan.
Reduce the heat to low, place a lid on the pan and cook the onions, checking on them and stirring every 10-15 minutes, for roughly 45 minutes to 1 hour.
Once onions are golden and starting to smell caramelized, remove from the heat.
For the mustard aioli:
While the onions are cooking, whisk together the mayonnaise, Dijon mustard, honey, garlic powder, salt and pepper in a small bowl. Cover with plastic wrap and place in the refrigerator until ready to use.
For the burgers:
In a medium bowl, mix together the ground beef, Worcestershire, garlic powder, salt and pepper. Form the beef into four 4 oz. patties and season the tops with additional Kosher salt.
Heat a non-stick flat grill pan over high heat and let warm up for approximately 3-5 minutes. Lightly coat the grill pan with canola oil spray.
Place the burger patties on the grill pan and cook 2 minutes on one side, then flip and cook another 2 minutes. Add jarlsberg cheese on top of all four patties then cover with a dome shaped lid to melt the cheese.
After about 20-30 seconds, the cheese should be melted. Remove the lids and transfer the burger patties to a cutting board using a spatula.
Once all burger patties are cooked, toast the buns to desired doneness.
To assemble the burgers, spoon a generous amount of the mustard aioli on the bottom bun. Next, top with the burger patty, followed by a handful of caramelized onions and arugula. If desired, spread additional mustard aioli on the top bun, then place on the burger and serve.
Recipe by sweet caroline's cooking at https://sweetcarolinescooking.com/2021/04/28/doctors-orders-caramelized-onion-jarlsberg-arugula-burger/