In a heavy bottomed large skillet over medium low heat, heat oil and add in minced yellow onions. Reduce the heat to low and let the onion cook until caramelized to a golden brown- stirring occasionally - about 40 Minutes.
Blend together the roasted garlic cloves, caramelized onions, Worcestershire, parsley, whole grain mustard, salt, and pepper until smooth. Set aside.
Caramelized Onion Burgers
Using your hands, smash at least 2 heaping tablespoons of the caramelized onion mixture into the ground hamburger until completely (and as evenly as possible) combined. Try not to overwork the meat, if possible
Divide the mixture into four even quarters and form them into patties If possible, allow the burgers to rest for 30 minutes before cooking so the flavors can marry and the burger doesn’t get tough.
Grill, turning at least once, until desired doneness is reached. Place one slice of cheese on top of each patty and let melt. Remove from heat and let rest for five minutes before assembling.
While grill is still hot, lightly toast the buns. Coat buns with desired amount of mayonnaise.
Place arugula on the bottom bun (so it sticks to the mayo and is weighted down by the burger), add the burger patty, and 1-2 strips of cooked bacon on top of the burger followed by the top bun. Serve.
If you have extra caramelized onion mixture leftover, freeze it in an ice cube tray for later. Just fill each compartment with the two heaping tablespoons. Once they are frozen, break them from the ice cube tray and store in a plastic bag. It will keep like this for up to 6 months.