In a wok or skillet, stir-fry the ginger and garlic in 2 tablespoons of oil until fragrant, about 30 seconds. Add scallions, carrots, and red pepper and stir-fry over high heat for 2 minutes.
In a bowl, combine the chicken broth, soy sauce, and sugar. Add the Napa cabbage and broth mixture. Bring to a boil and simmer 5 minutes, stirring occasionally, until the vegetables are soft. Add sesame oil, cool for at least 15 minutes, and strain. Fold in the minced shrimp.
Fill and roll the egg roll wrappers, using 1 tablespoon of filling for each roll. Working with 1 wonton wrapper at a time, place the wrapper with one corner of the diamond closest to you. Place 1 teaspoon of the filling in the center of the wrapper. Roll the corner closest to you over the filling. Brush the top corner with water. Fold in the sides of the wonton and continue rolling the egg roll up until it is closed. Press to seal, set aside, and continue with the remaining ingredients.
In a skillet set over moderately high heat, heat the remaining oil and saute the egg rolls until golden brown on all sides, using tongs to turn them. Serve when cool enough to eat, with dipping sauce.