Cook the rice using a little less water than usual. (Freshly cooked rice is best for gimbap.) Keep it warm until all the filling ingredients are ready.
Cut the beef into thin strips (about 1/2-inch thick). Mix well with the next 5 seasoning ingredients. Set aside.
Bring a large pot of water to a boil over high heat and prepare a cold water bath. Blanch the spinach, place immediately in cold water, then squeeze out excess water. Season with the sesame oil and salt.
Crack and beat the eggs, in a bowl with a spoon or a fork. Stir in a pinch of salt. Heat a small non-stick pan over medium-low heat. Add the eggs to the pan. When the bottom is set but the top is still runny, flip it over 1/3 way and again to make a thick omelette. Transfer to a cutting board and let it cool. Cut lengthwise into 1/2-inch thick strips. (You can make a thin, round egg omelette and use 2 pieces per each roll, if it's easier for you.)
Julienne the carrots. Heat a lightly oiled pan over medium-high heat. Stir-fry the carrots until softened. Lightly season with a pinch of salt. Transfer to a plate.
Cut the fish cake lengthwise into about 3/4-inch thick strips. Heat the pan with cooking oil (about 1 teaspoon) over medium heat. Stir fry the fish cake with the soy sauce, sugar, sesame oil until softened, about 2 minutes.
Stir-fry the beef over medium-high heat until cooked through, 2 - 3 minutes.
Arrange all the ingredients together on a plate.
While the rice is still hot/warm, add the sesame oil and salt. Mix well by lightly folding with a rice paddle or large spoon until evenly seasoned. Add more salt to taste if needed.
Put a seaweed sheet, shiny side down and longer side towards you, on a cutting board or a bamboo mat if available. Spread about 3/4 cup to 1 cup of rice evenly over the seaweed, using a rice paddle or your fingers preferably. Leave about 1.5 inch space at the top.
Lay the prepared ingredients on top of the rice, leaving some uncovered rice at the top for sealing.
Lift the entire bottom edge with both hands and roll over the filling away from you, tucking in the filling with your fingers. Put firm pressure over the roll with the help of the bamboo mat (or aluminium foil), if using, to close everything in tightly. Then, continue to roll again, putting pressure evenly over the roll using both hands.
Rub or brush the roll with a little bit of sesame oil for extra flavor and shiny look. Apply a little bit of sesame oil to a sharp knife. (This will keep rice from sticking to the knife. Repeat as necessary after each cut. Wipe the knife with a damp towel as needed.) Cut the roll into 1/2-inch thick bite sizes.
Notes
Each roll takes about 3/4 to 1 cup of cooked rice. For this recipe, I used 1.5 cups of raw rice for 4 rolls. If you cook 2 cups of rice, you can make 5 or 6 rolls.
You can adjust the number and the amount of the filling ingredients to your liking. Omit any ingredient(s) you don't want or have, and/or add any ingredients you like.
Cucumber is great as a substitute for spinach or as an addition. See my Vegan Kimbap for preparing cucumbers.