Rinse the rice several times until the water runs clear. Cook according to package directions. Let cool slightly and season with 1 tablespoon sesame oil and ½ teaspoon kosher salt. Mix gently and set aside.
Bring a pot of water to a boil. Blanch spinach for 30 seconds, then transfer to ice water. Squeeze out excess moisture and season lightly with a pinch of salt and a few drops of sesame oil.
Beat the eggs with ¼ teaspoon salt. Heat a lightly oiled skillet over medium heat and cook into a thin omelet. Let cool, then slice into thin strips.
Heat ½ tablespoon neutral oil in a skillet. Sauté carrots with a pinch of salt for 2–3 minutes until slightly tender. Remove and set aside.
In the same skillet, cook Spam strips until lightly browned. Add gochujang and toss until evenly coated. Remove from heat.
Add the fish cakes to the skillet and stir-fry with soy sauce and oligodang syrup for 2–3 minutes until glazed.
Place a sheet of nori shiny-side down on a bamboo rolling mat. Spread about ¾ cup seasoned rice evenly over the nori, leaving a 1-inch border at the top.
Arrange spinach, egg strips, carrots, Spam, fish cakes, pickled yellow radish, and burdock root across the center of the rice.
Using the bamboo mat, tightly roll the kimbap away from you, sealing the edge with a few grains of rice if needed.
Repeat with remaining ingredients and nori sheets.
Lightly brush the outside of each roll with sesame oil.
Slice each roll into bite-sized pieces with a sharp knife and serve.