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Lay chicken on cutting board. Using a meat mallet, pound each breast to about 1/4 inch thickness. Cut the green onions the length of the chicken breasts. Arrange 2 or so green onions along the edge of a chicken breast. Roll the chicken tightly around the onion and secure with kitchen twine. Repeat with remaining chicken and onions. Set aside.
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Combine soy sauce, sugar, and garlic in a zip top plastic bag. Add the chicken wraps and marinate for about an hour at room temperature.
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Preheat the oven to 400°F.
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Heat the vegetable oil in a large, oven proof skillet over medium-high heat - I use a cast iron skillet. Remove the chicken wraps from the marinade and place in the skillet, reserving the marinade. Cook, turning occasionally until well browned, 5 - 6 minutes. Pour the remaining marinade over the chicken and place the skillet in the oven and continue cooking an additional 5 - 6 minutes.
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Serve with additional soy sauce if desired. Fried rice is a great side dish with this meal.