Position an oven rack in the center of the oven and preheat to 400 degrees F. Combine the apples, onions, lemons, sage sprigs and chicken broth on a rimmed baking sheet.
Sprinkle the turkey generously on both sides with 2 tablespoons salt and 1 tablespoon pepper, and rub all over with the paprika. Put the turkey over the apple mixture skin-side up and rub the skin with the olive oil. Roast until the skin is golden brown and crisp, and an instant-read thermometer inserted into the thigh registers 155 degrees F, 1 hour to 1 hour 10 minutes. Transfer the turkey to a cutting board and tent it loosely with foil; let rest 30 minutes before carving. (The turkey will continue cooking as it rests.)
Drain the juices from the baking sheet through a mesh sieve into a small saucepan, pressing on the solids to extract as much liquid as possible. (Discard the solids.) Skim off any fat from the juices with a spoon if necessary, and keep warm over low heat.
Carve the turkey and arrange on a warm serving platter. Melt the butter in a medium skillet over medium heat, swirling the butter in the pan occasionally, until it becomes deep nutty brown, about 5 minutes. Add the sage leaves and cook, stirring, until the edges curl up, just a few seconds. Spoon the brown butter and sage leaves all over the turkey, and sprinkle with some of the flaky sea salt. Serve the pan juices on the side.
A spatchcocked turkey is one that has been split open and then flattened like a book. Ask the butcher to do it for you.
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