Sauce: Combine ingredients in a small sauce pan and cook over medium heat, stirring frequently. Transfer to a small serving bowl, let cool. (Microwave in a small serving bowl, much easier, but stir often).
Wontons: In a large skillet, heat 2 tablespoons of the oil over medium-high heat and saute mushrooms until soft and lightly browned. Stir in cashews, scallion, carrot, garlic, ginger, and soy, and cook 5 more minutes until fragrant. Set aside to cool slightly, then transfer to a food processor and pulse until cashews are finely ground and mixture is somewhat smooth.
Place 2 teaspoons os the mushroom mixture in the center of each wrapper and moisten the edges of the wrapper with water. As you fold the wrapper in half to enclose the filling, squeeze out as much air as possible to avoid air pockets. If you want to be fancy, fold little pleats along the curved side of the dumpling before pinching together to seal. Just make sure it is sealed so you don't get leaks.
Heat remaining 2 tablespoons oil in a large nonstick skillet over medium-high heat. When hot, arrange dumplings in the skillet without overcrowding them. Cook for a few minutes until the bottom sides are lightly browned, checking gently with a spatula.
Being VERY careful to avoid spattering oil, slowly add hot water to reach a depth of 1/2 inch. cover immediately and cook for about 5 minutes, or until all the water has evaporated. Flip the dumplings and cook another few minutes in the remaining oil, until the other side is lightly browned as well.
OR: For steamed dumplings, place in a steamer basket and cook, covered, over steam for 15 minutes. Check periodically and add more water to the bottom of the steamer if necessary.