Preheat oven to 425°F.
Toss apple, onion, and chicken with olive oil, salt, pepper, and sage and roast in a shallow baking pan for 20 minutes or until fairly browned. Let cool.
Place cooled contents into food processor bowl and pulse until coarsely chopped, then empty mixture into a bowl and mix with softened butter.
Place about 1 tablespoon of mixture into the center of wonton skin and bring up each of the 4 corners with your fingers pinching together so that contents become a nice purse dumpling. If necessary, dampen edges of wonton skin with water to help them stick.
Heat olive oil in a skillet at medium heat. (Do not use a non-stick pan.) Add 6 gyoza purses to oil in pan; cook until they stick to the bottom of the pan. Gently separate them from the pan with a spatula. Add ⅓ cup water to the pan and cover to steam dumplings. When the skins are translucent, the gyoza are done.
Remove them from the pan, lower temperature and add butter. Cook over low heat until butter begins to color and smell turns nutty. Add pecans and sauté for 1 minute, stirring constantly, until pecans are toasted. Pour a small amount over each cooked gyoza.