Season the chicken with salt and black pepper, place it in a covered skillet with the broth and bring it to a simmer.
Cook over medium-low heat for 30 to 35 minutes, turning once, until the thighs reach 165°F and pull apart easily.
Whisk the mayonnaise, vinegar, horseradish, Dijon and sugar until smooth, then measure 1/2 cup for the chicken, 1/4 cup for the cabbage and keep the remaining clean sauce for assembly.
Shred the hot chicken with 2 forks, discard excess pan liquid and toss the meat with 1/2 cup white sauce.
Toss the cabbage with its measured 1/4 cup white sauce and toast the potato buns cut-side down for 1 minute.
Divide the sauced chicken among the buns and top each with 1/2 cup dressed cabbage and a spoonful of reserved clean sauce.