Heat a large skillet over medium heat. Add the butter and let it melt. Toss in the sliced onions, seasoning with salt and pepper. Cook until golden brown and crispy, about 10-15 minutes. Remove from skillet and set aside.
In the same skillet, add the burger patties. Cook to desired doneness, about 4-5 minutes per side for medium. In the last minute, place a slice of blue cheese on each patty to melt.
Slice the burger buns in half and toast them lightly on a separate skillet or in the oven until golden brown.
Assemble the burgers by spreading mustard on the bottom half of each bun. Layer with arugula or kale, followed by the patty with melted blue cheese, fresh tomato slices, and crispy fried onions. Cap with the top half of the bun.
Serve immediately while hot and juicy, paired with your favorite sides.
