1. Rinse chicken and pat dry. Cut breast halves crosswise into 1/8-inch-wide strips.
2. In a 5- to 6-quart pan over high heat, bring about 4 quarts water to a boil; add noodles and cook, stirring occasionally, until tender to bite, about 3 minutes. Drain noodles and return to pan.
3. Meanwhile, pour oil into a 10- to 12-inch frying pan over medium-high heat. When hot, add onion, mushrooms, green onions, and garlic; stir often until onion and mushrooms begin to brown, about 8 minutes. Add chicken and stir often until no longer pink in the center (cut to test), about 3 minutes. Add rice wine, vinegar, and soy sauce; cook to blend flavors, 1 to 2 minutes longer.
4. Add chicken mixture to noodles in pan and stir over medium-high heat until hot, 2 to 3 minutes. Divide equally among four wide bowls.