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Step 1
For the Brine: In a large mixing bowl, prepare an ice bath. In a 2-quart saucepan, stir together 1 quart (1L) water with salt, brown sugar, peppercorns, and bay leaves over medium heat until brine reaches a gentle simmer and salt and sugar have dissolved. Set saucepan in ice bath, making sure it doesn't cause an overflow, and stir brine until cooled to room temperature. In a large zipper-lock bag, add turkey wings, shallot, thyme, and rosemary. Carefully pour cooled brine on top, seal, pushing out any excess air, and refrigerate for at least 3 and up to 8 hours.
Step 2
Drain wings, discarding brine and aromatics. Dry wings well with paper towels. Set aside.
Step 3
For the Braise: Preheat oven to 350°F (175°C). In a large Dutch oven, heat 1 tablespoon (15ml) oil over medium-high heat until shimmering. Working in batches, cook wings, turning, until golden brown on both sides, about 7 minutes. Transfer wings to a tray or platter and set aside.
Step 4
Lower heat to medium and add onion, carrot, and celery to Dutch oven and cook, stirring, until onions are translucent and carrots and celery have softened slightly, about 4 minutes. Transfer vegetables to a heatproof bowl or plate.
Step 5
Add remaining 2 1/2 tablespoons (35ml) oil to Dutch oven. Stir in flour and cook, stirring, until flour is a little darker than the color of peanut butter, about 10 minutes (the timing can vary here, so be sure to follow the color indication).
Step 6
Whisk in stock in 1/2-cup increments, whisking well between additions to ensure a smooth texture, until all the liquid has been added. Bring to a low boil, then whisk in sage, black pepper, granulated garlic, onion powder, and smoked paprika. Season very lightly with salt. Remove from heat.
Step 7
In a baking dish large enough to hold the wings in one layer, spread vegetables in an even layer. Arrange wings on top in an even layer, top side down, then pour the pan gravy on top until it comes halfway up the wings (this should be all of the gravy); if there's not enough, add enough additional stock to bring the liquid halfway up the wings.
Step 8
Cover baking dish tightly with aluminum foil, then bake, turning once halfway through so wings are top side up, until turkey wings are fork-tender, about 1 hours 15 minutes. Turn oven up to 375°F (190°C) and remove foil from baking dish. Return wings to oven and cook until gravy has thickened to a spoon-coating consistency and wings have darkened slightly, about 15 minutes longer. Season gravy with additional salt, if needed, and let rest 10 minutes.
Step 9
Serve smothered wings with their gravy on rice, mashed potatoes, or as desired.