In a 4- to 5-quart covered pan over high heat, bring broth to a boil, then reduce heat to low and keep hot. In a 6- to 8-quart covered pan over high heat, bring 3 quarts water to a boil. Meanwhile, in each of six to eight large, deep bowls, place about 1 teaspoon oyster sauce and 1/2 teaspoon sesame oil.
When water boils, add noodles. Stir to separate; cook just until barely tender to bite, about 2 minutes. Drain.
Add 3 to 4 quarts water to unwashed pan; cover and bring to a boil over high heat. Meanwhile, with chopsticks or tongs, divide noodles evenly among bowls; cover with foil to keep warm.
When water is boiling, add won ton; stir to separate and cook until filling is no longer pink in the center (cut to test), 3 to 4 minutes (6 minutes if frozen). With a large, flat wire strainer or slotted spoon, lift won ton out and divide evenly among bowls.
Ladle hot broth over won ton and noodles. Sprinkle servings with green onions and cilantro. Offer white pepper and more oyster sauce to add to taste.