Preheat oven to 375°F. Bring a large pot of water to boil over high heat.
While waiting for water to boil, heat 1 tbsp sesame oil in a large skillet over medium heat. Pat chicken thighs dry with a paper towel and sprinkle on both sides with salt and pepper. Place skin down in skillet and cook, undisturbed, until skin is well browned, about 5-7 minutes.
Place chicken skin-up on a sheet pan and bake for about 10 minutes, to internal temperature of 165°F.
Meanwhile, without wiping out the skillet, add savoy cabbage. Cook, stirring infrequently, until cabbage was wilted and is browned/charred in places, about 5-10 minutes. Remove from heat.
While cabbage and chicken are cooking, boil soba noodles per package directions. Drain and return to pot.
When chicken is done, remove to a cutting board. Remove skins and bones and cut chicken meat into bite-size chunks.
In a small bowl, whisk together oyster sauce, rice wine vinegar, and 2 tbsp sesame oil.
Combine chicken, cabbage, soba noodles, and sauce in the skillet. Toss together over low heat. Serve. Top with fermented hot sauce, if desired.