Combine brown sugar, mustard, salt and cayenne pepper in large non-metallic container. Gradually whisk in the water, thyme and garlic.
Remove giblets and neck from turkey. Discard or refrigerate for another use. You can use the turkey lifter while in the brine mixture only. (Do not use in the fryer.)
Place turkey in brine mixture. Cover and refrigerate overnight.
Pat turkey dry with paper towels. Place turkey, breast side up, in basket. Slowly lower basket into hot oil, being cautious of splattering oil.
Maintain oil temperature at about 350°F. Fry turkey for 3½ to 4 minutes per pound. Remove from oil to check doneness. Insert an instant-read thermometer into the thickest part of the breast, not touching bone. The temperature should read 170°F. Insert the thermometer into the thickest part of the thigh, not touching bone. The temperature should be 180°F.
Remove turkey from hot oil and drain on paper towels. Let rest for 15 minutes before carving.