Step 1

To prepare pickled onion: Toast crushed red pepper, peppercorns, garlic and bay leaf in a medium saucepan over medium-low heat until fragrant, 1 to 3 minutes. Add vinegar; bring to a boil over medium-high heat. Add sugar and stir until dissolved. Place onion in a medium non-reactive bowl and pour the hot liquid over it. Let cool to room temperature, about 1 hour. Cover and refrigerate for 4 hours and up to 1 week.

Step 2

To prepare sandwich: Combine mayonnaise, mustard and Worcestershire in a small bowl.

Step 3

Heat 1 tablespoon ghee in a large nonstick skillet over medium-high heat. Add 2 to 3 bread slices and cook until browned on one side only, 1 to 2 minutes, adjusting heat as necessary. Transfer to a plate. Repeat 2 more times with 2 tablespoons ghee and the remaining bread.

Step 4

Add mushrooms, pepper, salt and 1 tablespoon ghee to the pan. Cook, without stirring, until browned and crisp, 2 to 3 minutes. Flip the mushrooms and press with a spatula to flatten. Top with cheese and cook until it is melted, about 3 minutes more. Transfer to a plate.

Step 5

Add the remaining 1 tablespoon ghee to the pan over medium heat. Crack eggs into the pan, cover and cook to desired doneness, 2 to 4 minutes.

Step 6

Spread the sauce on the untoasted sides of the bread. Layer the pickled onion, mushrooms, eggs and lettuce between the bread slices.