In large saucepan over medium high heat, bring 6 cups water to boil. Add chicken breast, and return to boil. Reduce heat to low. Simmer, uncovered, about 15 minutes. Remove meat from broth, and set aside to cool.
Bring broth to boil again, and add noodles. Cook, stirring occasionally, 5 to 7 minutes. Drain, reserving broth for another use if desired. Rinse noodles under cold running water until chilled. Drain again, and transfer to serving bowl. Toss lightly with 1 teaspoon sesame oil.
Cut or pull chicken meat into fine shreds, discarding skin and bones. Set aside.
Combine tahini and 3 tablespoons water, stirring to blend. Add chili oil, soy sauce, vinegar, 1 tablespoon sesame oil, peanut oil ,and garlic. Mix well.
Arrange the chicken on top of noodles in serving dish. Spoon sauce over all.