Step 1: Heat a splash of vegetable oil in a skillet over medium heat. Add the peeled and deveined shrimp and sauté for 3-4 minutes until they turn pink and are cooked through. Remove from the skillet and chop into small pieces.
Step 2: In the same skillet, add a little more oil if needed. Sauté the minced garlic and ginger for about 30 seconds until fragrant. Add the shredded cabbage, grated carrot, and chopped green onions. Cook for 2-3 minutes, stirring, until the vegetables are tender-crisp.
Step 3: In a large mixing bowl, combine the chopped cooked shrimp, sautéed vegetables, soy sauce, sesame oil, salt, and pepper. Mix everything thoroughly to ensure the filling is well combined and seasoned.
Step 4: Lay an egg roll wrapper on a clean, dry surface with one corner pointing towards you (like a diamond). Place about 2 tablespoons of the shrimp filling horizontally across the lower middle section of the wrapper.
Step 5: Fold the bottom corner of the wrapper tightly over the filling. Then, fold in the left and right corners towards the center. Roll the wrapper tightly upwards towards the top corner, sealing the edge with a small dab of water to help it stick.
Step 6: In a deep skillet or a sturdy pot, heat about 2 inches of vegetable oil over medium-high heat until it reaches 350°F (175°C).
Step 7: Carefully place the assembled egg rolls into the hot oil, frying in batches to avoid overcrowding the pan. Fry for 3-4 minutes on each side, or until they are beautifully golden brown and crispy.
Step 8: Remove the fried egg rolls with tongs or a slotted spoon and place them on a plate lined with paper towels to drain excess oil. Serve hot with your favorite dipping sauces.