Heat a large cast iron skillet over medium-low heat.
Add butter and swirl to coat the pan. Add onions and cook, stirring frequently for 20-30 minutes or until they are a light golden brown color but not fully caramelized. Remove to a bowl and set aside.
Turn heat up to medium-high and add mushrooms. Cook until the mushrooms release their liquid and reabsorb it, stirring frequently. The timing varies on the freshness of your mushrooms but should take about 7-10 minutes.
Add onions back to pan and season with salt and pepper, to taste. Cook on medium-low heat for 25-30 minutes or until mushrooms and onions are a deep golden brown color but not burned. Stir frequently to keep from burning.
Burgers:
Preheat grill to 350°F.
To a medium bowl, add ground meat, worcestershire sauce, olive oil, garlic powder, onion powder, salt, and pepper.
Mix with your hands until spices are well-combined with the meat. Add in diced pieces of Swiss cheese and gently incorporate them into the meat.
Divide meat into 4 equal portions. Roll each piece of meat into a ball, then form into a patty by squishing it between the palms of your hands. Patties should be about 4.5-5" in diameter. Using your thumb, place a small dimple in the middle of each patty.
If using a pellet grill, open the flame broiler lever to cook directly over the flames. Carefully place burger patties on grill and cook for 4-5 minutes.
Flip and cook an additional 3-4 minutes. Remove to a foil-lined flat grill pan or small sheet pan.
Top burgers with mushrooms and onions, then top with cheese. You can save some of the mushrooms and onions out to place on top of the melted cheese too, if desired.
Place pan back on grill and cook for 4-5 minutes, using indirect heat (not directly over the flame).
Remove pan from grill when cheese is melted.
Spread mayo on buns if desired, then add burgers to the bottom bun. Drizzle with ketchup if desired and place top half of bun on top. Enjoy!
Notes
Thaw ground venison in a paper towel-lined bowl to soak up extra liquid. This helps remove off flavors and helps the burgers stick together better.
Slicing onions root to tip (julienned) keeps the vegetable fibers more intact which helps keep the pieces whole when caramelizing them and prevents them from turning to mush.
Caramelized mushrooms and onions can be made ahead of time and reheated when ready to use.
Keep your burger mix cold when forming patties. If it has warmed up while mixing, place it back in the fridge for 30 minutes before making your patties. This helps keep the patties intact and prevents them from falling apart.
The Swiss cheese in the burger patties can melt and make a mess of your grill. Cover the bottom of the grill with foil if you are concerned about a mess.