In a large stockpot, whisk the brown sugar with the mustard, salt, and cayenne. Gradually whisk in the water, then add the thyme and garlic. Add the turkey, cover, and brine in the refrigerator for 35 hours.
In a turkey fryer or an 18-quart or larger stockpot, bring the oil to 400°F; this can take up to an hour. Remove the turkey from the brine and pat it dry inside and out with paper towels. Transfer the turkey to a frying basket, breast side up.
Lower the turkey into the hot oil and fry for 3 minutes per pound, 36 minutes. Lift the basket out of the fryer and drain the turkey on a rack set over a rimmed baking sheet for 15 minutes. Carve and serve.
