Remove all traces of skin, fat, and tendons from the chicken legs. Put them in the water in a medium saucepan. Bring to a boil, then reduce heat to a simmer. Cook for 45 minutes, skimming off any foam that rises to the top.
Remove the chicken legs from the poaching liquid. Take the meat off the chicken legs, then hand shred it.
Toss the meat with the sesame oil, soy sauce, and chili paste. Divide into two serving bowls.

Bring the chicken broth to a boil. Add the garlic, scallions, and noodles. Cook the noodles until they are al dente, 2 to 3 minutes. Ladle the soup into the bowls over the chicken.