Line a plate with two paper towels. Place two small portobello mushroom caps on the paper towels, then cover with two more paper towels. Top with another plate.
Meanwhile, in a small bowl, mix together mayonnaise, ketchup, worcestershire sauce, garlic salt, and hot chili sauce.
Place a slice of Swiss cheese down on a counter top, then place a slice of American cheese on top, folding the corners of the American cheese in towards the center. Do the same with Swiss cheese, forming a ball shape.
Place the cheese ball on one mushroom cap and then sandwich the other on top of it, pressing down to make the mushroom edges touch. Take a piece of clear plastic wrap and wrap it around the mushroom burger, twisting the sides like a piece of candy to release all of the air. Tuck the plastic wrap under the mushroom and refrigerate for 10-15 minutes.
Put the egg, flour, and bread crumbs into 3 separate bowls.
Remove plastic and coat mushroom burger in flour, egg, then bread crumbs. Dip in egg wash and bread crumbs one more time.
Heat vegetable oil in a large pan until it is 375ºF (190ºC) and fry mushroom burger for 1-2 minutes or until golden brown.
Remove burger onto paper towel to drain excess oil and let cool 1-2 minutes.