In large food processor add cream cheese, Pepper Jack, garlic powder, onion powder, green onions and cooked chicken. Pulse just a few times; just enough to combine.
Place heaping teaspoon in the center of each wonton and pull up sides and pinch. Lightly moisten fingers with cold water and pinch again. Press down gently on plate to get their bottoms flat (so they will stand up). Keep extra wontons skins covered with lighlty moistened paper towel while you work. Repeat until all wontons are full. Cover filled wontons with lightly moistened paper towel while you are working and heating the oil. .
Heat about 1 inch of vegetable oil to 350 – 375 degrees in a heavy skillet. Working in small batches fry until golden brown; less than 1 minute. Use tongs to brown on both sides if they fall over. Drain on paper towels.
In small bowl mix soy sauce, water, mirin and sesame oil. Serve warm wontons with sauce.