Heat the oil in a large skillet over medium-low heat.
Add the onion slices and cook until just turning soft, about 5 minutes or so.
Add the mushrooms. Stir and cook until the mushrooms and onions are caramelized and get some brown color. This can take up to 10 minutes. If too much liquid accumulates in the pan it will boil the vegetables not brown them, so drain some off if needed.
Season with salt and pepper. Stir and remove from heat.
Fire up your grill for direct medium-high heat cooking. You can also use an outdoor or indoor griddle or a large skillet on a stove.
Form the beef into 4-equally sized balls or patties. Season well with salt and pepper.
Place the patties on grill. Smash down with a spatula or grill press.
Grill the burgers for 2-3 minutes per side.
Top the patties with cheese and grill until they've reached your desired doneness and the cheese is melted. Remove from the grill.
Toast the buns as desired.
Make the burgers by placing the patties on the bun bottoms.
Add the mushroom and onion mix followed by a good drizzle of the steak sauce.