Figure out how much oil you need to fill your fryer: Put the turkey in the empty fryer and cover with cold water. Remove the turkey; mark the line where the water is in the pot. Empty the water and dry the fryer and turkey well.
Mix the smoked paprika, garlic powder, onion powder, cayenne pepper, thyme, 1 tablespoon salt and 1 1/2 teaspoons black pepper in a bowl. Sprinkle some of the spice rub inside the cavity of the turkey. Separate the skin from the breast meat with your fingers, starting at the top of the breast and sliding to the right and left, then working down. Massage some of the rub onto the meat under the skin. Sprinkle the remaining rub on the turkey's skin. Place the turkey on a sheet tray and cover with plastic wrap. Refrigerate overnight or up to 24 hours so the flavors can marry.
Fill your fryer with peanut oil to the line you marked (do not fill more than three-quarters of the way). Preheat the oil to 400 degrees F; it will take about 1 hour.
Meanwhile, remove the turkey from the refrigerator and let it come to room temperature as your oil heats. Once the oil reaches 400 degrees F, very carefully lower the turkey into the hot oil. (Most turkey fryers come with a basket for the turkey that has hooks and a handle to lower and lift; otherwise, get yourself a long set of sturdy tongs to grip the inside cavity and breast and an industrial kitchen fork to hold the back side of the bird.) Make sure the oil maintains its temperature while frying. Fry the turkey until the skin is dark golden brown and crisp, or until the internal temperature of the breast reaches 155 degrees F on an instant-read thermometer, about 45 minutes. Carefully remove the turkey from the oil and let it rest and drain on a wire rack, about 30 minutes. Do not cover the turkey with foil or it will lose some of its crispness. The internal temperature will rise to 165 degrees F while resting. Transfer the turkey to a board or platter.
Photograph by David Malosh
Recipe courtesy Gina Neely for Food Network Magazine