For khao soi paste:
Place chiles in a small heatproof bowl, add boiling water to cover, and let soak until softened, 25-30 minutes.
Drain chiles, reserving soaking liquid. Purée chiles, shallots, garlic, ginger, cilantro stems, coriander, turmeric, curry powder, and 2 tablespoons soaking liquid in a food processor, adding more soaking liquid by tablespoonfuls, if needed, until smooth.
Heat oil in a large heavy pot over medium heat. Add khao soi paste; cook, stirring constantly, until slightly darkened, 4-6 minutes. Add coconut milk and broth. Bring to a boil; add chicken. Reduce heat and simmer until chicken is fork-tender, 20-25 minutes. Transfer chicken to a plate. Let cool slightly; shred meat.
Meanwhile, cook noodles according to package directions.
Add chicken, 3 tablespoons fish sauce, and sugar to soup. Season with salt or more fish sauce, if needed. Divide soup and noodles among bowls and serve with toppings.
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