Heat sesame oil in a large stockpot over medium-high heat. Add green onion whites, ginger, and garlic. Cook until ingredients are browned, about three minutes.
Add broth, bring to a boil, then reduce heat. Cover the pot and simmer for 10 minutes.
Ladle 1 cup of broth into a separate bowl. Slowly add the miso, stirring constantly until the miso is dissolved. Add mixture back into the broth.
Return soup to a simmer. Add chicken, carrots, and noodles. Continue to cook until the noodles are tender, about three minutes.
Remove pot from heat, add spinach, and stir until spinach begins to wilt.
Ladle soup into individual serving bowls. Garnish with remaining green onions.