Thai Chicken Soup

INGREDIENTS

• 2 vegetable oil
• 1 bunch scallions, whites and pale-greens thinly sliced, dark greens sliced 1-inch thick
• 4 garlic cloves
• 1 1-inch piece ginger, peeled, finely chopped
• 1 large carrot, peeled, shredded
• 1 jalapeño, with seeds, thinly sliced (or seeded if you prefer less heat)
• 1 quartered stemmed shiitake mushrooms
• 1 skinless, boneless chicken breasts or thighs
• 1 low-sodium chicken broth
• 1 14-ounce can coconut milk
• 1 (or more) fish sauce (such as nam pla or nuoc nam)
• Kosher salt
• 2 thinly sliced sugar snap peas
• 1 lime zest, divided
• Fresh tender herbs (such as cilantro or basil)
• Lime wedges (for serving)