Preheat the oven to 375 degrees F. Season the chicken with 1 teaspoon salt, and pepper to taste. Heat 1 tablespoon vegetable oil in a large ovenproof skillet over medium-high heat. Add the chicken, skin-side down, and cook until golden, about 5 minutes. Transfer the skillet to the oven (do not flip the chicken); roast until a thermometer inserted into the thickest part of the meat registers 165 degrees F, about 25 minutes. Transfer to a cutting board and let rest 5 minutes.
Meanwhile, make the dressing: Whisk the vinegar, mustard and honey in a large bowl. Gradually add the remaining 1/3 cup vegetable oil, whisking until emulsified.
Remove the chicken from the bone; slice. Thinly slice the apples and add to the dressing along with the walnuts, escarole and endive; season with salt and pepper. Toss the salad and top with the chicken and cheese. Serve with bread.