1. Attach food grinder attachment to electric stand mixer according to manufacturer's instructions. Grind chicken with garlic according to manufacturer's instructions using the fine grinding plate.
2. Cook chicken mixture in hot oils in a large nonstick skillet over medium heat 8 to 10 minutes or until chicken is done. Stir in green onions, next 3 ingredients, and 1 tablespoon lime juice. Cook 1 minute.
3. Grate ginger using the large holes of a box grater. Squeeze juice from grated ginger into a small bowl to equal 1/2 teaspoon; discard solids.
4. Stir together ginger juice, chili sauce, 1 teaspoon water, and remaining 1 teaspoon lime juice until blended. Serve chicken mixture on lettuce leaves with ginger-chili sauce. Garnish, if desired.
*Hot chili sauce may be substituted.