In a 5- to 6-quart pan over high heat, bring 2 1/2 to 3 quarts water to a boil. Add chicken and return to a boil; cover and remove from heat. Let stand until chicken is no longer pink in center of thickest part (cut to test), about 18 minutes. If still pink, return water to a simmer, cover, remove from heat, and let stand for a few more minutes. With tongs, lift chicken from water; let cool. Tear chicken into 1/2-inch-thick shreds.
Return water to a boil over high heat. Add noodles and cook, stirring occasionally, until barely tender to bite, 3 to 4 minutes. Drain, rinse with cold water until cool, and drain again thoroughly. Rinse and dry pan.
In the pan, over medium heat, stir sesame seeds in oil until golden, 2 to 4 minutes. Stir in garlic and remove from heat. Stir in vinegar, soy sauce, sugar, and cayenne. Spoon 3 tablespoons of the sesame vinaigrette into a small bowl and reserve. Add the noodles and half the carrots, celery, bean sprouts, and cilantro to pan; mix well.
Mound noodle mixture equally on dinner plates. Pour any dressing left in pan into bowl with reserved dressing. Top noodle mixture with remaining carrots, celery, and bean sprouts, then all the chicken. Sprinkle with remaining cilantro. Serve with reserved dressing to drizzle over chicken to taste.
Four types of noodles: Look for dried Asian noodles in the international section of the supermarket or in an Asian market.
Bean threads (saifun or cellophane noodles). Thin, wiry dried noodles, made from the starch of mung beans, turn clear and slippery when cooked in water or puffy and crisp when deep-fried. Neutral flavor.
Rice noodles (rice sticks, mai fun, mi fun). Dried white noodles, made from rice flour, vary from whisker-thin to about 1/4 inch wide. When cooked in water, they turn opaque and tender; when fried, they puff and crisp. Mild rice flavor.
Soba. Buckwheat and wheat flour infuse thin, tan Japanese noodles with robust, earthy flavor.
Wheat noodles (Chinese noodles or Oriental noodles, mein). Available in many forms, these all-purpose noodles taste similar to spaghetti and go by many names.