To prepare pot stickers, bring sherry and 1/2 cup water to a boil in a small saucepan. Remove from heat; add mushrooms. Cover and let stand 30 minutes or until mushrooms are tender. Drain and discard liquid. Finely chop mushrooms. Combine mushrooms, bok choy, and the next 9 ingredients (bok choy through chicken), stirring well.
Working with 1 wonton wrapper at a time (cover remaining wrappers with a damp towel to prevent drying), spoon about 1 1/2 teaspoons chicken mixture into center of each wrapper. Moisten edges of dough with water; bring 2 opposite corners to center, pinching points to seal. Bring the remaining 2 corners to center, pinching points to seal. Pinch 4 edges together to seal. Place pot stickers on a large baking sheet sprinkled with cornstarch.
Heat 2 tablespoons oil in a large nonstick skillet over medium-high heat. Add half of pot stickers; cook 2 minutes or until bottoms are golden brown. Carefully add 1/2 cup water to pan; cover and cook 4 minutes. Uncover and cook 3 minutes or until liquid evaporates. Repeat procedure with remaining 2 tablespoons oil, pot stickers, and 1/2 cup water.
To prepare sauce, combine mirin and the remaining ingredients. Serve with the pot stickers.