Step 1

For the mushroom caps, preheat the oven to 375° F. Put mushroom caps on a medium baking pan. Rub caps all over with 1/4 cup of canola oil, then season both sides with the salt and pepper. Arrange mushrooms gill side down in a single layer and roast until tender, 30 to 35 minutes. Remove from the oven and set aside to cool. Slice mushrooms in half horizontally (through their bellies, like you're slicing a hamburger bun).

Step 2

Put the 1 1/2 cups flour in a wide dish, beat 3 eggs in another wide dish, and put the panko in a third wide dish and set aside.

Step 3

For the filling, mix together the Muenster and cheddar cheeses, onion, garlic, remaining 1/4 teaspoon flour, remaining egg yolk, and cayenne pepper in a medium mixing bowl. Set aside.

Step 4

Shape the filling into 4 equal disks, 1/2 inch smaller in diameter than the mushroom caps. Place cheese disks on gill side of each mushroom cap, cover with other half of cap, cut side down, and gently press together to secure the filling.

Step 5

Dredge the stuffed mushroom caps in the flour, then dip in the beaten egg, coating them completely, then dredge in the panko. Set prepared mushrooms on a plate and refrigerate while heating the oil to deep-fry them.

Step 6

To cook the mushroom burgers, pour the oil into a deep, wide pot to a depth of 3 inches. Heat over medium heat until the oil reaches a temperature of 350° F on a candy thermometer. Deep-fry the prepared mushrooms, one or two at a time, until golden brown and crisp all over, about 3 or 4 minutes. Transfer mushrooms with a slotted spoon to drain on paper towels. Season with salt.

Step 7

Spoon the ShackSauce onto the top bun. Add a piece of lettuce and two slices of tomato. Transfer the mushroom burgers to the prepared buns. Enjoy!

Step 8

Whisk together all ingredients to combine.