Marinate the Beef: In a medium bowl, whisk together the soy sauce, brown sugar, sesame oil, minced garlic, grated ginger, grated pear/apple (if using), and chopped green onion until the sugar dissolves. Add the thinly sliced ribeye and sliced onion (if using). Toss to coat evenly. Cover and refrigerate for at least 30 minutes.

Cook the Bulgogi: Heat 1 tbsp neutral oil in a large skillet or wok over medium-high heat. Drain excess marinade from the beef. Add the beef in a single layer and cook for 2-3 minutes, stirring occasionally, until browned and just cooked through. Season lightly with black pepper. Remove from heat and set aside.

Fry the Eggs: In the same skillet, add a drizzle of oil over medium heat. Crack in the eggs and fry sunny-side up for 2-3 minutes until the whites are set but yolks are still soft. Season with a pinch of salt. Remove and set aside.

Toast the Bread: Spread softened butter on the cut sides of the rolls or bread slices. Heat a clean skillet or griddle over medium heat and toast the bread butter-side down for 1-2 minutes until golden and crispy.

Assemble the Sandwiches: Layer the hot bulgogi beef on the bottom half of each toasted roll/bread slice. Top with 2-3 slices of mozzarella per sandwich. Return to the skillet over low heat for 1 minute to melt the cheese. Slide a fried egg on top of the melted cheese. Close with the top half of the bread.