Place mushrooms in food processor; pulse 6 to 8 times to chop into small pieces.

Melt butter in skillet over medium heat. Add mushrooms; cook 3 to 5 minutes or until they begin to release their liquid. Add green onions, garlic, ginger, salt and pepper. Reduce heat to medium-low. Cook, stirring every 3 to 5 minutes, until mixture is dry and starting to brown, about 15 to 20 minutes.

Add wine and soy sauce. Increase heat to high and stir until all the liquid has evaporated, about 1 minute.

Transfer mushroom mixture to food processor. Add cheese and cilantro. Process until smooth. Spread mixture into a thin layer on a plate to cool. Chill in the refrigerator.

Lay 1 won ton wrapper at a time on work surface. Trim side edges with decorative cutting wheel, if desired. Place 1 rounded teaspoon of cooled mushroom filling in lower third center of wonton. Lightly moisten top and bottom edge with water using your finger or pastry brush. Fold bottom edge over filling and then fold over again to other side pressing gently around filling (to expel any air) and edges to seal. Pinch sides at filling to form shape.

Repeat with remaining wrappers and filling.* 

Heat peanut oil for deep frying to 350F.

Fry wontons, a few at a time, for 3 to 4 minutes or until golden brown.

Transfer to a cooling rack. Sprinkle with salt while still hot.

Arrange on platter with dipping sauce.