Gently stir together beef, 1/4 cup blue cheese, salt, black pepper, and worcestershire sauce just until incorporated.
Shape into 4 thin patties, and refrigerate.
For the onions:
Heat a cast iron skillet until hot over medium heat. Add the butter (or oil) to a large cast iron skillet, and add the onions.
Stir well, breaking up the onions and coating the onions in the butter.
Cook for 10 minutes. Stir and reduce heat to medium-low. Cook for another 25-35 minutes, stirring every 5-10 minutes, until golden brown. Remove the onions from the skillet and set aside.
Make the burgers:
Add the oil to the skillet and turn the heat to medium-high. Place the burgers in the hot skillet and cook for about 3-5 minutes until the burger easily releases.
Flip the burger and reduce heat to medium. Cook until the burger reaches desired doneness. (Note: I cooked mine about 5-8 minutes for a medium/medium-well burger)
Remove from heat and let rest a few minutes before serving
Serve each burger with about 1 tablespoon blue cheese, 1/4 cup onions and 1/4 cup arugula with or without a hamburger bun. Enjoy!
*Using lean (but not extra lean) ground beef keeps the amount of grease splattering to a minimum. I personally don't use extra lean ground beef because I like a little fat to give the meat flavor and tenderness.
Recipe from Living Well Kitchen https://blog.memeinge.com/blue-cheese-caramelized-onion-arugula-burger/