In a large skillet, heat 1-2 teaspoons of oil over medium heat. Add the minced garlic, ginger, and the white parts of the green onions. Sauté for 2-3 minutes until fragrant.
Stir in the shredded cabbage, grated carrot, and diced shiitake mushrooms. Cook for about 5-7 minutes until the vegetables are tender.
Add the chopped kimchi, grated tofu, soy sauce, and toasted sesame oil to the skillet. Mix everything well and cook for another 2-3 minutes. Remove from heat and let it cool slightly.
Fill a shallow dish with warm water. Dip one rice paper into the water for about 10-15 seconds until it becomes pliable.
Lay the softened rice paper flat on a clean surface. Spoon about 1-2 tablespoons of the filling onto the lower third of the rice paper. Fold the sides inward and roll it up tightly, like a burrito.
In the same skillet, add enough oil for frying (about 1/4 inch deep) and heat it over medium-high heat. Once the oil is hot, carefully place the dumplings in the skillet, seam side down. Fry until golden brown, about 3-4 minutes per side.
Remove the dumplings from the skillet and place them on a paper towel-lined plate to drain excess oil. Serve hot with a dipping sauce made from rice vinegar, gochujang, soy sauce, and toasted sesame oil.