Before you start chopping the veggies, you need to reconstitute the dried shitake mushrooms and remove the water content from the tofu.
To reconstitute the mushrooms, place them in a bowl and soak them in hot water for about 30 minutes. Since they float to the top, place another bowl over them to keep them submerged in the water. Set aside.
To extract as much water content from the tofu as possible, first remove it from the package and discard the water. Pat dry and place on a dish. Cover it with a dish towel or paper towels then place a heavy bowl on top of it. Set aside for 30 minutes.
While the tofu and mushrooms are doing their thing, chop up the other vegetables and set aside.
After 30 minutes, remove the mushrooms from the water. You can either discard the water or reserve part of the liquid (the non-gritty top part) to be used in your wonton broth if you're doing the soup version (gives it great shiitake flavor!) Squeeze the mushrooms to remove excess water, slice off their tough stems if necessary then chop them finely.
Next, drain the water that was extracted from the tofu and transfer to a large mixing bowl. Mash the tofu well with your hands. Add the chopped vegetables and the other ingredients to the mashed tofu and mix well.
Unwrap your package of wonton wrappers and place one sheet on a plate or clean surface. Put a small spoonful of filling in the center. Using a pastry brush or your finger, brush some of the beaten egg mixture along the edges then fold the wrapper over to form a triangle. Press down to seal firmly and remove air pockets if necessary. If they're not sealed tightly, you'll get lots of oil splatter when you fry them up and you don't want that. Make sure you don't over stuff the wontons either. I would say about a small teaspoonful fits perfectly.
Place folded wontons on a large plate, preferably in a single layer. If you need to stack them a little, make sure you don't stack them directly on top of each other because the centers have a tendency to stick after awhile.
When you're done with folding, pour about 1-1/2 inches of vegetable oil into a deep frying pan or wok and heat to about 300 degrees. You can check to see if the oil is hot enough by sticking the tip of a chopstick into the oil and seeing if the edges will bubble. If it does, it's ready for the wontons!
Fry the wontons until golden brown. It's important to control the heat when you're frying them. Keep it on medium heat and adjust as you go along as it's easy to burn the wontons if you're not careful. Flip them frequently so they brown evenly on both sides. Drain them on paper towels and serve immediately.
To prepare the sauce, whisk all ingredients in a bowl and serve as a dipping sauce or spoon it over the fried wontons.