Heat oil in a large heavy pot over medium-high heat. Cook scallions, garlic, and ginger, stirring frequently, until softened, about 4 minutes.
Add carrot, jalapeño, and shiitake mushrooms and cook for another 2-3 minutes.
Add chicken broth and bring to a simmer. Add chicken and cook until cooked through, about 10-15 minutes.
Stir in coconut milk and fish sauce. Bring to a simmer.
Add sugar snap peas and cook for 2-3 minutes, until tender-crisp.
Stir in half of the lime zest.
Serve immediately, garnished with fresh herbs and remaining lime zest, and lime wedges.