Preheat the oven to 425° F. Season the chicken generously with a little bit of salt and pepper. Heat 1 tablespoon of butter and the oil in a large, deep ovenproof skillet over medium-high heat. Dredge the chicken in flour, shaking off any excess, and lay it skin-side-down in the skillet. Cook the chicken, turning once, until it is golden brown on both sides, about 8 minutes. Transfer to a platter.
Reduce the heat to medium, add the leeks and celery seeds, and cook until slightly soft, about 4 minutes. Pull the skillet from the heat, add the vermouth, and use a wooden spoon to scrape up any browned bits that cling to the pan. Add the broth, apple cider, thyme, and the 1 teaspoon of salt and bring to a boil.
Meanwhile, use a fork to work the 2 tablespoons of flour into the remaining 2 tablespoons of butter. Whisk the butter mixture into the skillet until dissolved. Stir in the apples. Return the chicken (now skin-side up) along with any collected juices to the pan. Transfer to the oven and bake, uncovered, until the chicken is cooked through, tender and lightly browned, about 25 minutes.
Notes
*If preparing this recipe as gluten free, just be sure that you use a gluten free all-purpose flour in place of regular, and use chicken broth and apple cider that is free of gluten.
*Leeks have a reputation for being "gritty." To clean leeks, place in the basket of a salad spinner with water and swish leeks around to flush out the dirt. Then let the dirt settle to the bottom of the bowl and lift out the basket.
*Serve with hot cooked noodles that have been tossed with butter and parsley.