Mix all of the filling ingredients in a medium bowl.
Place approximately 1 tablespoon of the mixture in the middle of a wonton wrapper. Wipe some beaten egg along the inner edge and fold the wonton wrapper together, pressing firmly, to create a tight seal. Repeat with the remaining filling.
Heat a large nonstick skillet on medium high heat. Add the oil. When the oil is warm, add the potstickers to the skillet, careful not to overlap them (but you can crowd the pan). Cook until they are just starting to brown, approximately 2-3 minutes.
Add 1/4 cup of water and immediately cover with a tight lid. Cook until most of the water is evaporated, approximately 5-7 minutes. Remove the lid and let the remaining water cook off as the wrapper shrinks onto the filling, just a minute or two longer.
Repeat if necessary and serve warm with additional sauce for dipping.
Notes
You can find wonton wrappers in the produce section, typically near the fresh herbs.