In a 4-to-5-quart pot, combine the boneless skinless chicken thighs, sliced garlic, chopped ginger, scallion whites (set aside the scallion greens for later), kosher salt, freshly ground pepper, and water. Bring the mixture to a boil over high heat. Once boiling, reduce the heat to medium-low and allow it to simmer uncovered. Stir the pot occasionally and let the chicken cook until it 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. This will create a tangy sauce for drizzling over the soup later. I like to add a generous spoonful of chili oil for extra depth and heat, but you can adjust the amount based on your spice preference.
Once the chicken is cooked and tender (from Step 1), use tongs to remove the chicken pieces from the pot and transfer them onto a cutting board. Add the dried ramen noodles and carrot matchsticks to the simmering broth, cooking according to the noodle package instructions, usually about 3–4 minutes. Meanwhile, use two forks to shred the cooked chicken into bite-sized pieces.
Once the noodles are tender, return the shredded chicken (from Step 3) to the pot. Gently stir everything together and rewarm for about 1 minute. Taste the soup and adjust the seasoning with more salt or pepper if needed. Sometimes I like to add an extra dash of soy sauce at this stage to bump up the umami flavor.
Divide the hot soup, with noodles and chicken, among serving bowls. Top each bowl with a generous handful of reserved scallion greens for freshness and crunch. Drizzle each serving with about 1 tablespoon of the soy-vinegar sauce from Step 2, adding more to taste. Serve immediately, letting everyone add extra sauce or chili crisp as desired.