Shake Shacks Shroom Burger

Name: Shake Shack's 'Shroom Burger
Ingredients: For the mushroom caps and filling:; 4 4-inch portobello mushroom caps; 1/4 cup canola oil, plus more for deep-frying; 1/2 teaspoon kosher salt; 1/8 teaspoon freshly ground black pepper; 1 1/2 cups plus 1/4 teaspoon flour, divided; 3 eggs plus 1 egg yolk, divided; 2 cups panko breadcrumbs; 1 1/2 cups grated Muenster; 1/2 cup grated cheddar; 1/2 teaspoon minced onion; 1/8 teaspoon minced garlic; 1 dash Kosher salt; 4 tablespoons ShackSauce (below); 4 hamburger potato buns, toasted; 4 pieces green leaf lettuce; 8 1/4-inch slices ripe plum tomatoes; For the ShackSauce:; 1/2 cup Hellman's mayonnaise; 1 tablespoon Dijon mustard; 3/4 teaspoon Heinz ketchup; 1/4 teaspoon kosher dill pickling brine; 1 pinch cayenne pepper