Heat the sesame oil in a large pot over medium heat. Add the grated ginger, minced garlic, and sliced carrots. Cook for 3 minutes until the carrots begin to soften.
Pour in the chicken broth and bring the mixture to a simmer. Stir occasionally while it heats.
Place the miso paste in a small bowl and ladle in about half a cup of the hot broth. Whisk until smooth, then stir the mixture back into the pot.
Add the shredded rotisserie chicken, chili flakes, and baby spinach. Simmer for 5 minutes until the spinach wilts and the chicken is heated through.
Ladle the soup into bowls and top each serving with the sliced green onions.