Add the sesame oil and a pinch of salt to the warm cooked rice. Gently mix until every grain is evenly coated and seasoned. Set aside to cool slightly before assembling the kimbap.
Prepare the carrots by cutting them into thin matchsticks. Heat a pan over medium heat with a little oil, then saute the carrots with a pinch of salt for 2–3 minutes, or until they are slightly softened but still have a bit of crunch. Transfer to a plate and let them cool.
Crack the eggs into a bowl, add a pinch of salt, and whisk until well combined. Heat a lightly oiled non-stick pan over medium heat. Pour in half of the egg mixture, tilting the pan to create a thin, even layer. Once the bottom is set but the top is still slightly wet, use chopsticks or a spatula to gently roll the egg into a log. Repeat with the remaining egg mixture. Let the egg rolls cool slightly, then cut each roll in half lengthwise.
Peel the cucumber, then cut it in half lengthwise. Slice each half lengthwise again into long strips, then remove the seeds by running your knife along the center of each strip. This helps keep the kimbap from becoming watery.
In a small bowl, whisk together the soy sauce, sugar, minced garlic, grated ginger, black pepper, and sesame oil until the sugar has dissolved. Set aside.
Heat a pan over medium-high heat. Add the beef slices and cook until they are no longer pink. Pour in the prepared sauce and continue cooking, stirring frequently, until the beef is fully coated and most of the sauce has reduced and caramelized. Remove from the heat and let it cool slightly.
Place a sheet of roasted seaweed (nori), shiny side down, on a bamboo rolling mat. Spread an even layer of cooked rice over about 8/9 of the seaweed, leaving about 2 cm (¾ inch) uncovered at the top edge. Arrange a line of carrots, egg, cucumber, and cooked beef on top of the carrot, egg, and cucumber. Starting from the edge closest to you, tightly roll the kimbap away from you, using the bamboo mat to help compress the filling. Seal the roll by lightly pressing the uncovered edge against the roll.
Lightly brush the outside of the finished kimbap with sesame oil, then sprinkle with toasted sesame seeds for extra flavor and a beautiful finish.
Using a sharp knife, slice the kimbap into bite-sized pieces, wiping or lightly brushing the knife with sesame oil between cuts to prevent the rice from sticking.
Serve immediately.
