Melt the butter with the oil in a medium skillet over medium heat.
Add the remaining ingredients and cook, stirring often, for 20 minutes or until the mushrooms are golden and the onions are softened. Remove from heat.
For the burgers
Fire up your grill for indirect cooking.
Divide the beef into 6 2 1/2 ounce meatballs. Pack tightly so they stay together.
Take a shot glass and push it down into the middle of each ball, wrapping the meat around the outsides of the glass. Don't press down so far that you puncture through the bottom - you'll want about 1/3" of meat all around. Push the glass down gently onto a flat surface to flatten the bottom.
Season the patties with salt and pepper and place onto the grill over indirect heat. Grill until the meat is almost at the desired doneness, 45 minutes or longer.
Spoon in the mushroom topping and add the cheese. Continue cooking until the cheese is melted.