In a food processor, pulse the chicken until finely chopped.
Add the arrowroot powder or cornstarch, salt, ice water, and egg white to the chopped chicken in the food processor bowl. Process until a velvety paste develops, about 1 minute.
Heat a large saucepan over medium-high heat. When hot, add the oil, shallot and mushrooms. Cook, stirring frequently, until the shallots are slightly softened, 2 to 4 minutes.
Toss in the ginger and garlic and cook until fragrant, about 1 minute.
Pour in the broth. Increase the heat to high to bring the soup to a boil.
Turn the heat down to medium and add the coconut aminos or soy sauce and fish sauce to the soup. Taste and adjust for seasoning.
Add the chicken paste to the broth and, working quickly, use a spatula to break it up. Cook until the chicken is no longer pink, then stir in the spinach and cook until wilted, about 1 minute more.
Turn off the heat and stir in the toasted sesame oil. Taste and adjust the seasoning as needed and serve.