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Step 1
Using a vegetable peeler, remove the zest from the limes in strips; remove only the colored portion, not the white pith underneath. You will need a total of 8 strips, each 2 to 3 inches long. Juice the limes; set aside.
Step 2
In a large saucepan, combine 6 cups water, the zest strips, lemon grass, 2 halved chilies, shallots, garlic, ginger, Thai chili paste, 1 teaspoon salt and ¼ teaspoon white pepper. Add the chicken, cover and bring to a gentle simmer over medium-low; simmer until the thickest part of the breast reaches 160°F, 25 to 30 minutes. Using tongs or a slotted spoon, transfer the chicken to a medium bowl; set aside to cool. Remove and discard the solids from the broth.
Step 3
Return the broth, covered, to a simmer over medium. Add the mushrooms, re-cover and cook until tender, 10 to 15 minutes. Meanwhile, shred the chicken into bite-size pieces, discarding the skin and bones.
Step 4
When the mushrooms are tender, return the chicken and any accumulated juices to the pan. Stir in the sliced chilies, ⅓ cup of the reserved lime juice, the fish sauce and ¼ teaspoon white pepper. Taste and season with additional lime juice, fish sauce and pepper. Serve sprinkled with the cilantro.