In a 4-to-5-quart pot, combine the chicken thighs, sliced garlic, chopped ginger, scallion whites, kosher salt, pepper, and water. Bring to a boil over high heat. Reduce heat to medium-low and simmer uncovered, stirring occasionally, until the chicken is very tender and cooked through, about 15 minutes.

While the soup simmers, whisk together the black rice vinegar, soy sauce, and toasted sesame oil in a medium bowl. Stir in crispy chili oil to taste. Set aside for serving.

Using tongs, remove the cooked chicken from the broth and transfer to a cutting board. Add the dried noodles and carrot matchsticks to the simmering broth. Cook according to package directions, about 3 to 4 minutes. Meanwhile, shred the chicken into bite-sized pieces using two forks.

Once the noodles are just tender, return the shredded chicken to the pot. Stir gently and rewarm for about 1 minute. Taste and adjust seasoning with extra salt or a dash of soy sauce if needed.

Ladle the soup into bowls. Top generously with reserved scallion greens. Drizzle about 1 tablespoon of the soy-vinegar sauce over each bowl and serve immediately with extra chili crisp on the side.