Korean Omelet Sandwich

Mixing tomatoes, rice vinegar, salt, and a pinch of black pepper together in a bowl; cover with plastic wrap and refrigerate. Beat eggs, salt, and a pinch of black pepper together in a bowl. Heat 1 tablespoon oil in a large skillet over high; sauté cabbage, carrot, and onion in hot oil until the cabbage wilts, about 1 minute. Reduce heat to low and pour egg into the pan. Tilt pan to spread egg around the skillet; cook until the edges turn golden brown, 2 to 3 minutes and flip. Add more oil as necessary to prevent sticking and continue cooking until the bottom of the egg is golden brown 2 to 3 minutes more. Quarter the omelet. Top each piece with a slice of cheddar cheese and let it melt slightly. Melt 1 tablespoon butter in a separate skillet over low heat. Toast bread slices in the skillet and toast until browned on the bottom, 1 to 2 minutes per side. Flip the bread, add the remaining butter, and lift the bread to allow the butter to get under it; continue cooking until browned, 1 to 2 minutes more. Place each piece of omelet onto a toasted bread slice and sandwich with the remaining bread slices. Toss perilla leaves with the tomato salad; serve with the sandwiches.