Remove turkey pieces from the refrigerator and let them sit at room temperature for 30-45 minutes. Dry off with paper towels.
Preheat oven to 350° degrees. Rub the turkey pieces with the spice rub and olive oil, being sure to rub some under the skin of the breast half, then place them on a rack in a large roasting pan with at least 2-inch high sides (or separate them on two medium-sized racks and pans). Pour the broth or water in the bottom of the pan(s) and place in the oven. Roast for 30 minutes.
Remove pans from oven and baste the turkey parts with the marinade. Put them back in the oven and roast for another 15 minutes, then baste again. Let the turkey roast for another 15-30 minutes or until a thermometer inserted reads at least 165 degrees (being sure to not temp it against a bone).
When the meat is cooked through, remove from the oven and let sit at least 15-20 minutes before cutting into them. Carve the breast meat away from the ribs and breast bone, looking out for any small bones, and carve the leg meat off or serve the legs whole.
Notes: If the meat is getting too browned during roasting, loosely cover the turkey with foil. And if you need more servings, add another breast half (or whole breast) to the recipe and roast according to directions.