Cook the short-grain rice according to package directions until tender. While hot, transfer to a wide bowl and gently fold in a drizzle of sesame oil, a pinch of fine salt, and a sprinkle of toasted sesame seeds to season; let cool to room temperature.
Prepare fillings: Blanch the spinach briefly in boiling water, squeeze out excess water, then toss with a small amount of sesame oil and a pinch of salt.
Sauté the carrot julienne in a little vegetable oil with a pinch of salt just until tender-crisp; set aside to cool.
Whisk the egg with a pinch of salt, cook into a thin omelet in a lightly oiled pan, then roll and slice into strips.
Slice the eomuk fish cake into strips and sauté briefly with a splash of soy sauce, a pinch of sugar, and a little sesame oil until glazed and flavorful; remove from heat.
Lay a sheet of gim shiny-side down on a bamboo mat or clean surface. Spread an even layer of the seasoned rice over the gim, leaving a 1–2 cm border at the top edge.
Arrange a line of fillings across the center of the rice: spinach, carrot, egg strips, glazed fish cake, a couple of strips of danmuji, and some pickled burdock root for crunch and tang.
Roll tightly from the bottom using the mat to shape and compress the roll; press gently to seal the edge. Lightly brush the finished roll with a little sesame oil and sprinkle with toasted sesame seeds.
Using a sharp, slightly damp knife, slice the roll into 1–1.5 cm rounds. Serve immediately or pack for later—gimbap keeps well and is delicious at room temperature.
