Remove the skin and bones from chicken, then mince or finely chop it. Mix the chicken with the water chestnuts, green onions, 2 teaspoons of the vegetable oil, the cornstarch, salt, pepper and soy sauce.
Put 1/2 teaspoon of the chicken mixture in the center of each wonton skin. (Cover the remaining skins with a moist kitchen towel to keep them pliable.)
Fold the bottom corner of the wonton skin over the filling to the opposite corner, to make a triangle. Brush the right corner of the triangle with egg.
Bring the corners together below filling the, pressing out any air, and pinch the left corner to the right corner to seal. Repeat with the remaining filling and wonton wrappers. Cover the filled wontons with a dampened towel or plastic wrap to stop them drying out.
Heat 1 1/2 inches of oil in a wok (or use your deep-fryer) to 350 degrees F (180 degrees C). Fry 8 to 10 wontons at a time until they are crispy and golden brown, turning them a few times.
They will take about 3 minutes to cook fully. Drain the cooked wontons on paper towels and serve with sweet and sour sauce, Chinese lemon sauce or another tasty dipping sauce.