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Cook noodles according to package directions, omitting salt and fat. Drain and keep warm.
Cut chicken into 1-inch pieces. Place chicken breast halves in a shallow dish. Combine 1 tablespoon flour, 1/4 teaspoon salt, and 1/8 teaspoon pepper, stirring well with a whisk. Sprinkle flour mixture over chicken; toss to coat.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add chicken to pan; sauté 4 minutes or until browned. Remove chicken from pan. Add remaining 1 tablespoon oil to pan. Add garlic, tarragon, and mushrooms to pan; sauté for 3 minutes or until liquid evaporates and mushrooms darken. Add white wine to pan; cook 1 minute. Stir in remaining 1 tablespoon flour; cook 1 minute, stirring constantly. Stir in broth, remaining 1/4 teaspoon salt, and remaining 1/8 teaspoon pepper; cook 1 minute or until slightly thick, stirring frequently.
Return chicken to the pan. Cover and simmer 2 minutes. Uncover; cook 1 minute or until chicken is done. Stir in noodles; cook 1 minute or until thoroughly heated. Place about 1 1/2 cups chicken mixture on each of 4 plates; top each serving with 1 tablespoon cheese.