Place the hot water, kosher salt, and brown sugar into a 5-gallon upright drink cooler, and stir until the salt and sugar dissolve completely. Add the ice and stir until the mixture is cool. Gently lower the turkey into the container. If necessary, weigh down the bird to ensure that it is fully immersed in the brine. Cover and set in a cool dry place for 8 to 16 hours.
Remove the turkey from the brine, rinse, and pat dry. Allow to sit at room temperature for at least 30 minutes prior to cooking.
Place the oil into a 28- to 30-quart pot and set over high heat on an outside propane burner with a sturdy structure. Bring the temperature of the oil to 250ºF. Once the temperature has reached 250ºF, slowly lower the bird into the oil and raise the temperature to 350ºF. Once it has reached 350ºF, lower the heat in order to maintain temperature. After 35 minutes, check the temperature of the turkey using a probe thermometer. Once the breast reaches 151ºF, gently remove from the oil and allow to rest for a minimum of 30 minutes prior to carving. The bird will reach an internal temperature of 161ºF due to carry-over cooking. Carve as desired.