Make the onion and mushroom mixture: Heat butter and oil in a large skillet to medium. Add onions and sauté, stirring every 5 minutes, for 20 minutes until the onions Begin to caramelize. Add mushrooms and cook for 10 minutes more. Add vinegar and brown sugar or honey, stir to combine and cook for another 4-5 minutes, until the vinegar reduces and most of the water has cooked out of the mushrooms. Season with salt and set aside.

Make the burger patties: combine all burger patty ingredients in a mixing bowl and mix gently with your (clean) hands. Be careful not to overwork the meat mixture as it may become tough. Form the mixture into 4 patties and set on a plate to rest while you get the grill ready.

Preheat your grill to medium and oil with a paper towel dipped in vegetable oil. Cook the burger patties for 5 minutes on each side; the last couple minutes of cooking add the cheese and a scoop of the onion-mushroom mixture so the toppings heat up. You can also grill the buns, cut side down on the grill, for about 30 seconds to toast them up at this point. The burgers should be at an internal temperature of 160 F when they are ready. Once cooked, remove from heat and let rest for at least 5 minutes before assembling the burgers.

To assemble the burgers, place a patty on the bottom side of each bun. Spread some mayo on the top of each bun, add a couple of lettuce leaves, cover each burger and serve.