Preheat the oven to 375° F.
Place the mushroom caps on a medium baking pan. Rub the caps with 1/4 cup of canola oil, and then season both sides with salt and pepper. Arrange the mushrooms gill side down in a single layer and roast until tender, for about 30 to 35 minutes.
Remove mushroom caps from the oven and set aside to cool. Slice them in half horizontally (through their bellies, like you're slicing a hamburger bun).
Grab 3 separate wide dishes. Put the 1 1/2 cups flour in one dish, beat 3 eggs in the second dish, put the panko in a third dish, and set aside.
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. Then set aside.
Shape the filling into 4 equal disks, 1/2 inch smaller in diameter than the mushroom caps. Place these cheese disks on the gill side of each mushroom cap, cover the disk with the other half of the mushroom cap, (cut side down) and gently press together to secure the filling.
Dredge the stuffed mushroom caps in the flour, then dip them in the beaten egg, coating them completely. Next, dredge them in the panko. Set the prepared mushrooms on a plate and refrigerate while heating the oil to deep-fry them.
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. Deep-fry the prepared mushrooms, one or two at a time, until golden brown and crisp all over. This should take about 3 or 4 minutes. Transfer the mushrooms to drain on paper towels, and season with salt.