How to Make It
Pound the Thai chiles, shallot, garlic and cilantro roots (or stems, if using) together in a mortar until bruised (or pulse 3 to 4 times in a food processor).
In a small saucepan, bring water to boil and season with 1/2 teaspoon of salt. Add the quinoa and simmer until the quinoa is just tender, 15 to 20 minutes. Drain the quinoa.
Meanwhile, in a large saucepan set over high heat, bring the stock and coconut milk to a boil, then reduce to a simmer. (The coconut milk will separate, which is fine.) Add the chile-garlic mixture and 3/4 teaspoon salt and cook, stirring occasionally, for 15 minutes.
Using a handheld immersion blender or jar blender, purée the soup until smooth. Strain through a fine-mesh sieve, pressing on the solids. Return the soup to the saucepan. Bring to a simmer and add the drained quinoa, brown sugar, fish sauce and lime juice. Adjust the seasonings, if needed. Divide the soup among four bowls and garnish with cilantro leaves. Serve immediately.