1. Preheat oven to 425 degrees F. Season chicken generously with some salt and pepper. Heat 1 tablespoon of the butter and the oil in a large, deep ovenproof skillet over medium-high heat. Dredge chicken in flour, shaking off any excess, and lay skin side down in the skillet. Cook chicken, turning once, until golden brown on both sides, about 8 minutes. Transfer to a platter.
2. 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 broth, apple cider, thyme, and the 1 teaspoon salt and bring to a boil.
3. Meanwhile, use a fork to work the 2 tablespoons flour into the remaining 2 tablespoons 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. Serve with Parslied Egg Noodles, if desired.
Bring a large pot of cold water to a boil over high heat and salt it generously. Add the noodles and cook, stirring occasionally, until al dente, about 5 minutes. Ladle 1/4 cup of the noodle cooking water into a medium skillet. Whisk in the butter bit by bit over low heat, letting each piece melt completely before adding the next, to make a creamy sauce. Stir in the parsley and season with salt and pepper. Drain the noodles, toss with the sauce, and serve immediately.
Know-How: Leeks have a well-deserved reputation for being gritty. To clean, split leeks lengthwise, then cut crosswise into half circles. Put cut leeks in a salad spinner with water (or set the basket in a large bowl of water if your spinner doesn't hold water). Swish the leeks to flush out the dirt, then let it settle to the bottom of the bowl. Lift the basket to leave all the grit behind in the water, drain, and spin-dry if needed.
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