For the Shack Sauce:

Whisk together mayonnaise, mustard, ketchup, white vinegar, and cayenne pepper in a small bowl. Refrigerate until ready to use.

For the Stuffing:

Preheat the oven to 180℃. Marinate the shiitake mushrooms with salt, pepper and oil. Place the mushrooms (stem side facing down) on a baking tray and bake for 15 to 20 minutes. Set aside until completely cool.
Slice shiitake mushrooms in half horizontally, in order to get two even slices without holes. Set aside.

For the Mushroom Filling:

Chop the trimmings of baked shiitake mushrooms and transfer them to a bowl.
Add Muenster cheese, cheddar cheese, onion powder, garlic powder, cayenne pepper and salt. Divide the filling into 4 equal portions.
Place a portion of the cheese mixture on the stem side of each sliced mushroom cap, leaving a 1/2-inch border. Top each with the other side of the mushroom, pressing firmly around the edges to seal the fiulling inside. If it's difficult to seal mushroom patties, remove some cheese mixture.
Meanwhile, heat sufficient oil for frying on a medium-low flame.
Take refined flour in a wide dish. Take the cornflour slurry in another wide bowl, place panko and bread crumbs in a third wide dish, set aside.
Dredge the stuffed mushroom caps in the flour, then dip in the slurry completely and then dredge in the panko.
Repeat the process with the remaining stuffed mushrooms.
Deep fry the mushrooms until crispy & golden brown, drain on absorbent paper.

For Assembling:

Cut the burger buns in halves. Add a spoonful of the prepared shack sauce on the bottom of the burger bun and spread evenly.
Place the fried mushroom burger patty on top, place a lettuce leaf and further add a slice of tomato. Add a generous serving of the shack sauce and cover with the top of the burger buns. Serve immediately with a side of potato wedges or wafers.