Mix the water and cucumber in a pitcher. Refrigerate for 1 hour.
Cook the rice according to the directions on the package. Once tender, remove from the heat and fluff with a fork. Season with salt and pepper to taste. Set aside.
In a nonstick skillet, melt butter over medium-low heat. Crack the eggs into the skillet. When the egg white is mostly set, fold the white over the yolk.
Dust both sides of the egg with flour using a 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 half of the rice in the center of the nori. Make sure the rice is warm but not hot. If rice is steaming hot, nori will likely tear.
Place a slice of cheese, avocado slices, mayonnaise, cooked egg, and remaining rice on top.
Carefully wrap the nori into a square using rice as glue. Flip it upside down and cut it in half. Wetting a knife with water before cutting can prevent the rice from sticking to the blade.
Serve egg gimbap with carrots, hummus, and cucumber water.