Season the chicken with salt and pepper. In a large, heavy saucepan, melt 2 tablespoons of butter with olive oil over medium-high heat. Add the chicken skin-side down. Brown on all sides. Transfer to a plate.
Pour off most of the fat from the pan. Add the diced onions, carrots, thyme and bay leaf, and cook until the onions are translucent, about 5 minutes. Add the Calvados, warm
slightly, and then remove the pan from the heat, stand back and ignite it with a kitchen torch to burn off the alcohol. Once the flames die, return it to the stove, add the cider, scraping up any brown bits. Bring to a boil and reduce by half. Add the stock and return the chicken to the pan. Simmer, covered, for 20 minutes. Transfer the breast pieces to a bowl. Cook the legs and thighs for 10 more minutes and add to the bowl. Keep warm.
Melt 2 tablespoons butter in a large skillet. Add the pearl onions and a pinch of salt, cover and cook over medium heat for 10 minutes. Uncover, add 1 tablespoon butter, and increase heat to medium-high. Place the apples in the center of the pan. Sear on each side for 10 to 15 minutes, until caramelized.
Strain the Calvados sauce and return it to the pan. Add the juices from the chicken. Whisk in the yogurt. Simmer until the sauce coats the back of a spoon. Season. Add the chicken pieces and warm through.