In a 4-to-5-quart pot, combine the boneless skinless chicken thighs, sliced garlic, chopped ginger, scallion whites, kosher salt, pepper, and 10 cups of water. Bring the mixture to a boil over high heat. Once boiling, reduce the heat to medium-low and simmer uncovered for about 15 minutes, or until the chicken is tender and cooked through.

While the soup is simmering, whisk together the black rice vinegar, soy sauce, and toasted sesame oil in a medium bowl. Add crispy chili oil to your preferred spice level.

Use tongs to remove the cooked chicken from the pot and move it to a cutting board. Add the dried ramen noodles and carrot matchsticks to the simmering broth. Cook according to the noodle package instructions, which is typically 3 to 4 minutes. While the noodles cook, use two forks to shred the chicken into bite-sized pieces.

Return the shredded chicken to the pot once the noodles are tender. Stir everything together and rewarm for approximately 1 minute. Taste the broth and adjust the salt, pepper, or soy sauce if needed to enhance the umami flavor.

Divide the hot soup, noodles, and chicken into serving bowls. Top each bowl with a generous amount of the reserved scallion greens. Drizzle each serving with about 1 tablespoon of the soy-vinegar sauce and serve immediately.