Position knife blade in food processor bowl; add butter, 1/4 cup cilantro, 2 teaspoons gingerroot, and garlic. Process until smooth. Shape mixture into a log. Wrap in wax paper, and chill until firm.
Place tuna steaks in a shallow dish.
Combine remaining 2 tablespoons gingerroot, brown sugar, and next 6 ingredients in a medium bowl; pour over tuna steaks. Cover and marinate in refrigerator 30 minutes, turning once.
Remove tuna steaks from marinade, reserving marinade.
Place grill rack on grill over medium-hot coals (350 to 400).
Place tuna on rack; grill, covered, 4 to 6 minutes on each side or until fish flakes easily when tested with a fork, basting occasionally with reserved marinade.
Transfer tuna to serving plates.
Divide chilled butter mixture into 6 pieces; place 1 piece of butter mixture on each tuna steak.
