Cook the rice for the sushi according to the directions on the package. Once tender, remove from the heat and set aside to cool to room temperature. Season with salt and pepper to taste. Set aside to cool down.

Marinate the beef: In a bowl, mix soy sauce, brown sugar, rice wine vinegar, sesame oil, garlic, and scallions. Add the beef and massage the marinade into the meat. Set aside for 30 minutes or longer if preferred.

Heat olive oil in a pan over medium heat. Sauté the onions until translucent. Remove beef from the marinade and set marinade aside. Add the beef in batches and cook until lightly browned on both sides. Avoid overcrowding to prevent steaming.

Add the marinade and bring to a simmer until the sauce slightly thickens. Set aside.

Make the Sushi: In a nonstick skillet, melt butter over medium-low heat. Crack the eggs into the skillet. When the egg white is mostly set, flip the egg over. Dust both sides of the egg with flour using a fine mesh strainer or sieve. Pour a mixture of soy sauce and maple syrup over the egg and cook for an additional minute. Set the egg aside.

Place a nori sheet on a cutting board. Spread the rice evenly on the nori in the center of the nori covering as much of it as possible. (Leave about 1 inch along the top and bottom of the sheet uncovered, as this will be how you seal your roll. Make sure the rice is warm but not hot. If rice is steaming hot, nori will likely tear.)

Arrange the cheese halves, avocado slices, mayonnaise, and cooked egg. (Cut the egg in half so it distributes evenly.)

Place the nori onto a bamboo sushi mat (or use a piece of plastic wrap on top of a towel).

Slowly start to roll the mat over the ingredients while gently pressing along the length of the "log." This motion presses the rice and the filling together and prevents the roll from being too loose. Continue rolling until you've rolled past the far edge of the seaweed sheet and the roll is sealed.

Once rolled, gently press and form the length of the roll to make sure it's compact and tight.

Slice the sushi roll into 6-8 pieces (you may have to rinse or wipe the blade in between slices to prevent sticking and tearing).

Pour sparkling water into a glass, add lemon slices and serve with ice if desired.

Serve sushi with kimchi, beef bulgogi, and lemon sparkling water.