Light your grill and set up for two-zone cooking. One zone at medium-high direct heat, the other for low heat.
Using your hands, combine the venison meat with the beef tallow, Worcestershire sauce and half of the kosher salt, being careful not to overwork the mixture.
Form the mixture into four equal sized patties and season on both sides with remaining salt and pepper.
Place the patties directly over heat. Allow them to cook for three minutes, flip the burgers and cook for an additional three minutes.
Move the burgers over to the low heat side of the grill and top each patty with an onion ring. Inside the onion ring, place one slice of cheese, (you may need to fold the cheese in order for it to fit inside the onion ring) an equal portion of sautéed mushrooms, then another slice of cheese.
Close the lid and allow the cheese to melt and the burger to finish cooking for 5-7 minutes, until the internal temperature of the burger is 140°F as indicated with a reliable quick read thermometer.
Remove from the grill, place on a brioche bun and serve hot.