In a large stockpot, boil a gallon of water. Add the brown sugar and salt, then stir to dissolve them. Add 1 gallon of cold water and let the brine cool to room temperature. Immerse the turkey in the brine and refrigerate it for 24 hours or overnight. Remove the bird from the brine, rinse it off, and pat it dry.
Fill a turkey fryer halfway with oil and heat the oil until it reaches 350 degrees. Gently lower the turkey into the oil and fry it for about 45 minutes, or until it reaches an internal temperature of 155 degrees (the temperature will rise to the USDA-recommended 165 degrees while the turkey rests). As the turkey cooks, do not let the oil get above 350 degrees. Remove the turkey from the fryer, tent it with foil, and let it rest for 20 to 30 minutes.