Season the patties all over with the BBQ rub and a small sprinkle of salt and pepper. Place in a dish large enough to hold the patties, and sprinkle a few splashes of Worcestershire sauce over both sides of all patties. Cover and place in the refrigerator.
MAKE THE MUSHROOM SAUCE
Heat the oil over medium-high heat in a large skillet or saucepan. Add the butter and swirl the pan until it is melted.
Add the onions and cook until soft and translucent, about 3 to 5 minutes.
Add the chopped thyme and cook until fragrant, about 30 seconds.
Add the mushrooms and ½ tsp of salt and ¼ tsp black pepper. Stir to fully coat the mushrooms.
Cook until mushrooms are soft and beginning to exude liquid. Add 1 tbsp Worcestershire sauce and the red wine (¼ cup) and cook until the liquid has reduced by about half, about another 4 to 5 minutes.
Sprinkle the flour all over the mushrooms and stir until fully incorporated. Cook for 2 minutes, stirring occasionally.
Add the beef stock and cook, stirring frequently, until thickened, about 4 to 5 minutes. Stir in the cream. Taste and add more salt, if desired. Keep warm on low heat on the stove.
FINISH OFF THE BURGERS
While the sauce is coming together, fire up your grill to medium-high heat.
Place the burgers directly over the heat and grill until desired doneness, about 5 minutes per side for medium to medium-rare.
Just before the burgers are done, place two slices of cheese on each patty. Also at this time, place the buttered side of all of the buns onto the grill.
Remove the burgers from the grill once the cheese has melted. Remove the buns once they are nicely toasted and brown around the edges.
Place burgers on the bottom buns and top with the warmed mushroom sauce. Top with the remaining buns.
These are 1/2 lb burgers. You can easily adjust the recipe to make smaller 1/4 burgers. You'll need to adjust the grilling time accordingly. We love using ground Wagyu beef, which can be found in the meat department of most well-stocked supermarkets. Grass-fed ground sirloin is an excellent choice, too. We go with 85% lean for the right amount of juiciness for the burger.Use your favorite types of mushrooms. We typically go with white button mushrooms that we slice nice and thick. However, shitake, portabella, or other types of wild mushrooms are great choices. If using dried, be sure to follow instructions on the packaging for re-hydrating. The mushroom sauce can be made up to 2 days in advance and then re-heated on the stove before serving. We often keep the mushroom sauce in a small Crock-Pot on the 'Warm" setting when serving a crowd.