Make the sesame sauce: In a food processor, combine the peanut butter, water, sesame oil, vinegar, ginger, garlic, and salt. Pulse until mostly smooth, then stir in the sesame seeds and red pepper flakes.
Make the Noodles: Next, bring a pot of water to a boil and cook the soba noodles according to the package directions. Drain and set aside in a large bowl.
Heat a large skillet with a little oil over medium heat. Add the chicken and sprinkle with salt and pepper. Sauté until barely golden brown on the outside. Add about one-third of the sesame sauce to the chicken with a little water to thin it out. It might splatter, so turn down the heat at this point and have a lid ready. Let the chicken finish cooking in the sauce for a few minutes. Remove the chicken from the pan, let cool slightly, and shred.
Add the remaining sauce, cucumber, cilantro, fermented carrots, and shredded chicken to the noodles and toss to combine. Serve warm or cold.