Copycat Shake Shack Shroom Burger

Name: Copycat Shake Shack Shroom Burger
Ingredients: 4 (4) portobello mushrooms, stems and gills removed; 1/2 c. all-purpose flour; 1 c. shredded Muenster; 1 c. shredded cheddar; 1/4 tsp. garlic powder; 1/2 tsp. kosher salt, divided; 1/2 tsp. freshly ground black pepper; 3 large eggs, divided; 2 c. panko bread crumbs; Vegetable oil, for frying; 4 potato buns, toasted; 2 plum tomatoes, sliced; 4 leaves green leaf lettuce; 2 tbsp. mayonnaise; 1 tsp. Dijon mustard; 1 tsp. ketchup; 1 tsp. hot sauce