Place fresh crimini mushrooms in the bowl of food processor and pulse for 15-20 seconds. The final result should resemble the texture and consistency of ground meat.
In a medium sized mixing bowl, add ground beef, ground crimini mushrooms, ½ tsp. salt, and pepper.
Using your hands, mix ingredients together to combine.
Form four good size burger patties and place on a plate to the side.
Heat 3 tbsp. of butter in a cast iron skillet and add thinly sliced onions. Cook over medium heat for 4-5 minutes, stirring occasionally to prevent burning. Add remaining salt and cook down for an additional 5 minutes until the onions are caramelized. Deglaze the pan with balsamic vinegar and cook for 1 minute.
Remove to a separate plate and add in remaining 1 tbsp. of butter. Once melted, cook burgers over medium heat until desired doneness. About 4-5 minutes per side for medium.
Spread ¼ of the Boursin cheese mixture on the bottom of each burger bun.
Top with ¼ of the balsamic onions, burger patty, and ¼ of the fresh baby arugula.
Finish with burger bun top and enjoy!