Heat a wok or large saute pan over medium-high heat, then add the vegetable oil. Once the oil is hot but not smoking, add the mushrooms and saute for 1 minute. Add the carrots and saute for 1 minute. Add the cabbage and saute for 1 minute. Add the bean sprouts and green onions and saute for 1 minute. Add the garlic and ginger and saute for 30 seconds. Add the soy sauce and sesame oil, stir to combine, and cook for 1 minute.
Transfer filling mixture to a sheet pan, spread into an even layer, and cool in the refrigerator, about 15 minutes. Drain any excess moisture from the pan after cooling.
Add the water to a small bowl and have a brush ready. Make sure to cover the wrappers with a damp paper towel to keep them from drying out.
One a cutting board, place one wrapper with the dusted cornstarch-side facing up and a corner pointed towards you. Add 3 tablespoons of filling to the lower third closest to you, making a 4-inch long mound. Tightly roll from the bottom corner towards the top, until about halfway up.Press on the edges of the filling to make sure it's packed and even. Brush a small amount of water on the left and right corners of the wrapper. Tightly fold the corners of the wrapper towards the center so it looks like an opened envelope.Brush the top corner of the wrapper with water. Tightly roll until it's completely sealed. Each roll should be about 4 inches long and about 1 ¼-inch wide. Repeat the process with the remaining egg rolls.
In a large pot, add enough peanut oil to fill about 3-inches in depth. Heat oil over medium heat until it reaches 350°F (177°C), this temperature should be maintained throughout frying.
Working in batches, add 3 to 4 egg rolls at a time into the hot oil. Occasionally move them for even cooking to achieve a crisp texture and golden brown surface, about 5 to 6 minutes.
Transfer egg rolls to a paper towel-lined baking sheet with a cooling rack on top. Repeat the cooking process with additional egg rolls.
Serve immediately with desired dipping sauce.
Notes
Freezing uncooked egg rolls: Freeze in a single layer on a parchment paper-lined sheet pan and then transfer to a resealable plastic bag or airtight container for up to 3 months. Fry directly from the freezer until golden brown, about 5 to 7 minutes.
Making a larger batch: For 20 egg rolls, cook double the amount of filling and use 20 wrappers.
Using thin spring roll wrappers: Add 2 tablespoons of filling and spread into a 4-inch long mound. Brush the wrapper with egg whites for better sticking. Fry until golden brown and crispy, about 2 to 3 minutes. The thinner wrapper cooks faster. Makes about 15 egg rolls.