Chef Kuniko Yagi of Los Angeles's Hinoki & the Bird makes a killer broth here—it's so good, you can eat it by itself. Slideshow: Chicken Soup Recipes
In a large pot, combine the chicken parts with the corn cobs, scallions, ginger, garlic, 2 1/2 teaspoons of salt and 4 quarts of water and bring just to a boil. Simmer over moderately low heat, skimming the surface, until the stock is flavorful, about 2 hours.
Meanwhile, in a large saucepan of salted boiling water, cook the vermicelli until al dente, about 2 minutes. Drain and cool under running water. Transfer the vermicelli to a bowl and toss with the oil to prevent sticking.
Strain the stock into the large saucepan. You should have about 2 1/2 quarts of stock; add water if needed.
Add the chicken breasts to the saucepan and return the stock just to a boil. Remove the saucepan from the heat, cover and let stand until the chicken is just cooked through, about 20 minutes. Using tongs, transfer the chicken to a plate; let cool slightly, then shred.
Add the corn kernels to the stock and bring just to a boil. Simmer over moderate heat until the corn is crisp-tender, about 3 minutes. Stir in the shredded chicken and the fish sauce and season with salt. To serve, fill bowls with the vermicelli and ladle the soup on top. Garnish with basil leaves and Thai chiles and serve.