Heat olive oil in a large saucepan or dutch oven over medium heat.
Add onion and carrot and sautee until slightly softened, 3 to 4 minutes.
Add mushrooms and cook until soft and shrunken in size, about 3 minutes more.
Add garlic, ginger, and chilies and stir until fragrant, about 1 minute.
Add chicken broth and water and bring to a simmer.
Add chicken breasts to liquid and simmer for 10 to 15 minutes or until chicken is cooked through (thickest part should read 165 degrees F on an instant read thermometer).
Remove chicken from pot and transfer to a plate to cool slightly before shredding with a fork.
Add noodles to simmering broth mixture and cook for 7 to 8 minutes or until just al-dente.
Add shredded chicken back to pot and cook for 1 to 2 minutes until heated through.
Stir in soy sauce to taste.
Divide among serving bowls and top with chopped cilantro.
