Place the turkey in a large food-safe plastic bag inside of an ice chest large enough to contain it. To make the brine, in a large bowl, whisk the brown sugar, mustard, salt, and cayenne pepper until smooth. Gradually whisk in the water, followed by the thyme and garlic and pour around the turkey in the plastic bag, submerging the turkey in the brine. Tie the bag closed, pack the cooler the rest of the way with ice, and brine for 24 hours.
Remove the turkey from the brine and pat dry inside and out with paper towels. Transfer the bird, breast-side up, to a turkey frying basket.
Place the turkey in the frying basket in a 30-quart stockpot and add
enough oil to barely cover the bird. Remove the turkey (in the frying
basket) from the stockpot and bring the oil to 400°F; this can take up
to 1 hour.
Carefully lower the turkey into the hot oil and fry for 3 minutes per pound, about 30 minutes. Lift the turkey in the basket from the fryer and drain over a draining rack for 15 minutes.
Remove the turkey from the basket, carve, and serve.