In a 4-to-5-quart pot, combine chicken thighs, sliced garlic, chopped ginger, scallion whites (reserve greens), kosher salt, pepper, and water. Bring to boil over high heat. Reduce heat to medium-low and simmer uncovered, stirring occasionally, until chicken reaches 165°F internal temperature and is cooked through, about 15 minutes.

During the first 5 minutes of simmering, whisk together black rice vinegar, soy sauce, and toasted sesame oil in a medium bowl. Stir in crispy chili oil to taste.

Remove cooked chicken from pot using tongs and transfer to cutting board. Immediately add dried ramen noodles and carrot matchsticks to simmering broth, cooking for 3 minutes until tender. While the noodles cook, use two forks to shred chicken into bite-sized pieces.

Return shredded chicken to pot. Gently stir and rewarm for about 1 minute. Taste and adjust seasoning with more salt or pepper if needed.

Divide hot soup with noodles and chicken among serving bowls. Top each bowl with reserved scallion greens. Drizzle each serving with about 1 tablespoon soy-vinegar sauce. Serve immediately.