Mix the sauce ingredients together in a small bowl or measuring cup and set aside
Run the mushrooms through a food processor until finely chopped (or you can do it by hand if you’re more patient than I am). Set aside.
Repeat with the drained water chestnuts.
Slice the scallions thinly (both green and white parts). Set aside.
Heat about a tablespoon of oil in a wok or large frying pan over medium-high heat. Add the coleslaw mix and stir-fry for about 2 minutes, or until the cabbage just begins to wilt.
Add the mushrooms with the sauce to the pan, combine well, and stir-fry for another 2 minutes.
Turn off the heat and add the water chestnuts and scallions. Stir to combine well. Allow filling to cool to room temperature.
For each egg roll, lay out one of the egg roll wrappers so that it appears diamond-shaped when facing you. Brush some of the beaten egg along the edges of the upper half of the diamond.
Place about 2 heaping tablespoons of the filling on the lower half of the diamond, below the two widest points.
Take the point closest to you, bring it over the filling, and tuck it under the filling, tautly. Roll up to the widest points, fold them in over the ends of package and continue to roll all the way up. Make sure that the outer edges are sealed firmly. Repeat with each egg roll.
If you have a deep-fryer, heat the oil to 350º F and fry the egg rolls, no more than 4 at a time, until they’re golden brown, turning once. This won’t take more than 2-3 minutes, so look sharp! I don’t have a deep-fryer, so I used my large frying pan, into which I poured about 2 inches of oil, heated to 350º F and proceeded the same way.
Serve with Chinese hot mustard, duck sauce, or your favorite dipping sauces.