Make the Schezuan dry-brined turkey: Place the coriander, fennel and Schezuan peppercorns in a small skillet over medium heat and toast until fragrant, about 5 minutes. Let the mixture cool, then transfer to a spice grinder and pulse until the spices are coarsely ground. Transfer the spice mix to a small bowl and stir in the salt; set aside.
Place the turkey on a clean work surface and trim off any excess pockets of fat or skin around the turkey. Flip the turkey so it is laying breast-side up. Using the heels of your hands, press down on the center of both breasts until you hear a cracking sound and the bird flattens.
Pat the bird very dry using a kitchen towel and rub the spice mixture on both sides. Transfer the turkey, breast-side up, to a wire-rack placed over a parchment-lined sheet tray. Tuck the wings underneath. Store the turkey in the refrigerator for at least overnight and up to 24 hours.
Remove the turkey an hour before roasting to take the chill off.
Place the reserved backbone and neck in a roasting pan with the carrots, onions, thyme, bay leaf and stock. Place the turkey on a wire rack.
Preheat the oven to 425°. Set one rack in the middle of the oven and another one below for the roasting pan. Place the turkey on the center rack of the oven. Place the roasting pan on the rack below the turkey and roast both for 30 minutes. Reduce the heat to 350° and finish roasting until the turkey is a deep golden brown and crisp or until an instant-read thermometer inserted into the thickest part of the thigh reads 165°, another 40 to 50 minutes. Remove the turkey from the oven and, using two grilling spatulas, transfer the bird to a platter. Let the turkey rest for 30 minutes before carving.
Meanwhile, make the gravy: Transfer the roasted neck-and-backbone turkey mixture along with the stock to a medium saucepan. Bring the liquid to a boil and reduce it by half, 10 minutes. Strain the stock through a fine sieve and discard the solids. Skim the stock of any excess fat before placing it back into the saucepan. Bring to a simmer.
Meanwhile, in a small saucepan over medium heat, melt the butter. When the butter begins to foam, whisk in the flour. Reduce the heat to low and whisk until the roux is golden brown, 2 to 3 minutes.`
Add 1 to 2 cups of hot stock to the roux, whisking constantly. Continue until all the stock has been added. Simmer until the gravy has thickened and the flavors have melded, about 5 minutes. Stir in the cream and season the gravy with salt and pepper. Carve the turkey and serve with the warm gravy.
Remove zest from lemons and limes and place zest in a medium bowl. Juice citrus, and reserve juice and citrus shells separately. Add butter, adobo sauce, chile powder, oregano and salt to the bowl. Beat with an electric mixer until combined. Slowly pour in about half of the citrus juice and beat until smooth.
Preheat the oven to 350°F. Carefully loosen skin over turkey breast and thighs; rub some of the butter mixture between meat and skin. Rub remaining butter mixture over skin and inside cavity. Place reserved citrus shells in cavity. Tuck wing tips under bird and tie legs together with kitchen string. Place breast-side up on a rack set in a large roasting pan and pour 3 cups of the broth and remaining citrus juice into the pan.
Roast until skin begins to deeply brown, about 30 minutes, then loosely cover with foil. Roast, basting turkey with pan juices about every 30 minutes, until an instant-read thermometer inserted into the thickest part of thigh, without touching bone, registers 165°F, 2 1/2 to 3 hours more. Transfer turkey to a cutting board and let rest for at least 30 minutes before serving.
To make the gravy, spoon off excess fat from pan juices, and place the pan on the stovetop over medium-low heat (over two burners if possible). Sprinkle in flour and cook, stirring, for 2 minutes. Stir in remaining 1 cup broth and cook, stirring, until slightly thickened. Strain gravy if desired. Carve turkey and serve with gravy.
Brine the turkey: In a large pot, combine the brine ingredients and place over medium-high heat. Bring to a simmer, stirring occasionally, until the sugar and salt have dissolved, 8 to 10 minutes. Remove from the heat and chill, uncovered, in the fridge until cool, about 1 hour. Make ahead: Once chilled, the brine can be covered and stored in the refrigerator for up to 1 week.
Rinse the turkey and neck, and place inside a 5-gallon container (or a large cooler). Pour the chilled brine over the turkey and top with a heavy plate, submerging the turkey. Cover and refrigerate for 6 to 8 hours
In a medium bowl, combine the paprika marinade ingredients. Make ahead: The marinade can be made and stored in the fridge for up to 5 days.
Remove the turkey from the brine and rinse under cold water. Discard the brine. Rinse out the 5-gallon container and return the turkey to it. Pour the paprika marinade over the turkey, rubbing all over the outside and the inner cavity; cover and chill in refrigerator for 8 to 10 hours.
Adjust the rack to the lower third of the oven, and preheat it to 400°
Cut off the turkey's wing tips, reserving them, and tie the legs together with kitchen twine. Combine the turkey neck, wing tips, red onions, carrots and bay leaves in a large roasting pan. Place the turkey on top, breast-side up, and rub with the salt and ¼ cup of the marinade.
Roast the turkey in the preheated oven until the skin is browned, 40 to 50 minutes, basting the bird and rotating the pan halfway through roasting, covering any dark spots with aluminum foil.
Lower the oven to 300°. Baste the turkey with marinade and pour 1 cup of water into the pan, creating steam to help cook the turkey. Continue to roast the turkey, rotating the pan and basting every 30 minutes, until a meat thermometer inserted into the thickest part of the thigh reaches just under 160°, 1½ to 2 hours. Remove the turkey from the oven and let it rest in the roasting pan for 30 minutes before carving. Serve with the pan drippings and roasted vegetables.
