Smash ginger root and lemongrass using a mallet to release flavors. In a large soup pot or dutch oven, combine: cilantro, ginger, lemongrass, onion, garlic, sugar, and coconut milk.
Bring to a simmer (medium heat), cover, and cook for 15 minutes. While the broth is cooking, sauté chicken until just underdone (will finish cooking once added to broth.)
After 15 minutes is up, strain the broth to remove all of the large chunks of herbs and vegetables. If needed, use a spoon to squeeze out all the juice.
Add juice from 3 limes and 1TB fish sauce to broth. Return broth to large pot and add sliced mushrooms. Cook on low heat for 4-5 minutes until mushrooms are soft.