Rinse, pat dry, and lightly pound the chicken. Slice it into three or four thin pieces. Season the chicken with salt and pepper.
Heat 2 Tbs. of the olive oil in a skillet over medium heat until it shimmers. Add the chicken slices and brown them on both sides, until just barely cooked throughout.
Add remaining olive oil to the skillet and cook onions and mushrooms over medium heat, adding just a little chicken stock as needed to keep mixture from sticking.
Cut chicken into quarter-inch dice. Add it to the pan, along with the peppers and spinach. Cook until the peppers have softened and the chicken has merged into the sauce. Add salt and pepper to taste.
Brush a wonton wrapper with chicken stock and place a tablespoon of stuffing in the center. Fold over the wonton and seal the edges with a fork to make dumplings.
Pour enough vegetable oil into a skillet to make it about a quarter-inch deep. Heat until almost smoking. Fry the dumplings for about a minute on each side, until lightly browned. Drain on paper towels.
Heat goat cheese, cream cheese and soy sauce in a microwave oven at 30% power, stirring after a minute, until it's warm and smooth. Thin the mixture with chicken stock until it has the consistency of a dipping sauce.