Preheat oven to 450 degrees, with racks in upper and lower thirds. Line one large rimmed baking sheet with aluminum foil; place a wire rack over a second rimmed baking sheet. In a large bowl, combine chicken, buttermilk, 1 teaspoon salt, and 1/2 teaspoon pepper; set aside (or refrigerate up to 2 days).
In another large bowl, whisk together flour, paprika, chopped rosemary, 1 tablespoon salt, and 2 teaspoons pepper. Lift chicken from buttermilk, a few pieces at a time (allowing excess to drip off), and dredge in flour mixture (shaking off excess); transfer to foil-lined sheet. Dredge chicken in flour mixture a second time; return to sheet.
In a 12-inch cast-iron skillet (or a 5-quart Dutch oven), heat oil and rosemary sprigs over medium-high. When rosemary sizzles rapidly, remove and discard. In three batches, fry chicken until golden brown, 4 to 5 minutes per side; transfer to prepared rack. (If chicken browns too quickly, reduce heat.)
Remove foil; set a rack on that baking sheet. Transfer half the chicken to this empty rack. Bake chicken on both racks until an instant-read thermometer inserted in thickest part of a thigh registers 165, 15 to 20 minutes. Let cool 5 minutes before serving.
You don't need a thermometer for the oil -- once the rosemary sprigs are hissing, it's time to add the chicken. Frying time is cut in half by finishing the chicken in the oven.