1-1/2 lb boneless chicken breasts (about 3 large)
1 head garlic, halved crosswise
2 bay leaves
4 whole allspice
1 ½ tsp kosher salt, plus more
8 oz. shiitake mushrooms, torn into bite size pieces
1 fresh chile, thinly sliced
1-1” piece ginger, peeled, finely chopped
1 tbsp. distilled white vinegar
1 tbsp soy sauce
fresh ground black pepper
sliced scallions and cilantro sprigs (for serving)
Place chicken, garlic, bay leaves, allspice, and 1 ½ tsp salt in medium pot, Cover with 6 cups water and bring to a bare simmer over high heat. Immediately reduce heat to medium-low, cover pot, and cook 8 minutes. Remove chicken from liquid and let cool slightly, then shred into bite-size pieces.
Strain stock through a fine mesh sieve into a clean pot; discard solids. Add mushrooms, chile, ginger, vinegar, and soy sauce to stock. Bring to a boil; reduce heat and simmer, stirring occasionally, until broth tastes rich and flavorful, 8-10 minutes. Season with salt and pepper, then add shredded chicken and simmer until meat is warm through. Divide soup among bowls and serve topped with scallions and cilantro.
DO AHEAD: Chicken can be poached 2 days ahead. Let chicken and broth cool separately. Wrap up chicken and transfer broth to an airtight container; chill.