Heat oil in a large saucepan over medium heat until shimmering. Add lemongrass, shallots, and cilantro sprigs. Stir and cook until softened but not browned, about 2-5 minutes. Stir in chicken broth, 1 can of coconut milk, and 1 tablespoon fish sauce. Bring to a simmer, then reduce heat to low, cover, and cook for 10 minutes to meld the flavors.
Strain the broth through a fine-mesh strainer into a large bowl or container, discarding the solids. Return the strained broth to the saucepan. Stir in the remaining 1 can of coconut milk, 2 tablespoons fish sauce, and sugar. Simmer over medium-high heat.
Stir in mushrooms and cook until just tender, about 2-3 minutes. Add the chicken and cook, stirring occasionally, until no longer pink, about 1-3 minutes.
Remove the soup from heat. Stir in lime juice, red curry paste, and Thai chiles. Taste and adjust seasoning with additional fish sauce, if needed. Serve hot, garnished with scallions, cilantro leaves, and lime wedges.