1. Preheat the oven to 400 degrees Fahrenheit.
2. In a microwave-safe bowl with a tight-fitting lid, place the Jerusalem artichokes and add one inch of water. Seal it tightly with the lid and microwave until the artichokes are cooked but still firm, seven minutes. Remove, uncover, and drain the artichokes.
3. In a medium bowl, whisk the cornstarch with a few tablespoons of the broth to dissolve, then whisk in the rest of the broth with the apple juice, vinegar, honey, mustard, and one teaspoon of the salt.
4. Season the chicken on both sides with the remaining one teaspoon salt. In the largest heavy skillet you have (ideally 12 inches, but 10 will do too), heat the oil over medium heat. Add the chicken, skin side down, and cook until the fat is rendered and the skin is golden and crisp, four to five minutes. Flip and cook until the underside is lightly golden, three to four minutes. Remove to a plate, leaving any rendered fat and juices in the pan. Add the onions and Jerusalem artichokes and cook, stirring, until the onions are lightly golden and the artichokes take on a little color, six to seven minutes. Add the sage, garlic, and chili and cook, stirring, one to two more minutes.
5. Whisk the reserved broth mixture to re-blend, add it to the skillet with the artichokes, and bring to a boil over medium-high heat, then reduce the heat to medium-low and simmer until the liquid thickens, five to eight minutes. Nestle the chicken, skin side up, and apples into the pan.
6. Carefully transfer the skillet to the oven and roast until the apples and Jerusalem artichokes are softened, the chicken skin is crisped and deep golden, and the sauce has reduced by about half, 35 to 40 minutes.
7. Garnish with more sage, then divide the chicken and sauce among plates.