Heat 1 tablespoon neutral oil in a 4-quart pot over medium heat until hot.
Add the onion wedges, ginger slices, turmeric, bay leaves, garlic, coriander, and black peppercorns. Sauté for 2 to 3 minutes until fragrant.
Place the chicken on top and season with 3 teaspoons kosher salt.
Cover the chicken with 8 cups of water. Bring to a boil. Reduce the heat to low and cover the pot. Simmer for 1 hour or until the thickest part of the chicken breast reaches 165ºF.
Shred the chicken:
Remove the chicken from the pot and transfer it to a large bowl. Once the chicken is cool enough to handle, pick the meat from the bones and shred it with your fingers. Reserve the carcass for stock, if you like.
Strain the broth:
Strain the broth into a large bowl through a fine-mesh sieve. Discard the spent aromatics.
Finish the soup:
Wipe out the pot and heat the remaining 1 tablespoon of neutral oil over medium heat. Add the sliced shiitake mushrooms and sauté until browned, about 5 minutes. Add the minced green onions, thinly sliced onions, and carrots. Sauté for an additional 3 minutes. Add the shredded chicken and cook for 1 minute.
Pour the reserved broth into the pot and bring to a boil. Reduce heat to low and simmer for 20 minutes or until the carrots are fork-tender. Taste and add more salt if needed.
Prepare noodles:
As the soup simmers, cook your favorite noodles according to package directions. Drain and divide the noodles between bowls.
To serve:
Ladle the soup over cooked noodles and serve with reserved minced scallion greens. Add a drizzle of chili oil on top if you like. Enjoy!