In a medium bowl or food processor, add the wine (or stock), soy sauce, balsamic vinegar, miso paste, and garlic, and whisk/blend well.
Place the mushrooms and onion slices in one (or more) shallow baking dishes or plastic food containers, and pour the marinade over them. (See notes.) Marinate the portobellos and onion slices for 30 minutes, turning over halfway through.
To make the cheddar sauce:
While the mushrooms and onions are marinating, make cheddar sauce. In a blender, combine the soaked cashews, veggie stock, apple cider vinegar, nutritional yeast, tomato paste, Worcestershire sauce, smoked paprika, salt, onion powder, garlic powder, and process on high until smooth and creamy.
To make the burgers
Set the oven to broil-HI, and position a rack about 4-5 inches from the flame. On a baking sheet or broiler pan, spread out the marinated mushrooms--top side up--and onions. Broil on high for 10 - 15 minutes (depending on the size of your mushrooms) or until cooked through and well browned. OR GRILL ON HIGH HEAT. Turn the mushrooms and onions over about half way through cooking. Spoon on a little leftover marinade after flipping, if desired. (See notes.)
Build your burger and enjoy. I like to turn my mushroom upside down--so that the it holds more of the cheese sauce!
If your mushrooms are large, you may have to use multiple dishes for marinating. Just be sure the dishes aren't too big, so that there's about 1/2 of liquid in the bottom of each dish.
You don't need any oil or a non-stick pan--the mushroom and onions won't stick.
You can broil the mushrooms for as little as 7 minutes (more if they're very large) and they'll be 'cooked.' However, I like the flavor of mushrooms when they're well done, I so recommend a longer cooking time.