Place the chicken breast in a saucepan and pour in enough chicken stock to barely cover it. Cover the pan and bring to a gentle simmer over medium-low heat. Once the liquid begins to simmer, slightly vent the lid and cook for 5 minutes. Remove the saucepan from the heat, cover tightly, and allow the chicken to rest in the hot liquid for 10 minutes. Transfer the chicken to a plate and shred it into thin pieces once cool enough to handle.

Add the remaining chicken stock to the pot and bring it to a simmer. Stir in the chopped ginger, lime juice, fish sauce, chili sauce, and brown sugar. Simmer gently for about 10 minutes to allow the flavors to infuse. Adjust seasoning if necessary, adding more chili sauce for spice or lime juice for brightness.

In a separate large pot, bring water to a boil. Add the soba noodles and cook for 6 to 8 minutes, or according to the package directions, until tender. Drain well. If not using immediately, rinse under cold water and toss with a small amount of olive oil to prevent sticking.

Return the broth to a simmer and add the bok choy and scallions. Cook for approximately 1 to 2 minutes, or until the greens are just tender and bright.

Divide the soba noodles among four serving bowls. Top each with the shredded chicken and ladle the hot broth and vegetables over the top. Garnish with fresh basil, mint, or cilantro, a sprinkle of red pepper flakes, and serve with lime wedges on the side.