Combine fish and 1/4 cup plus 2 tablespoons lime juice in a 13- x 9- x 2-inch baking dish. Cover and let stand 30 minutes.
Sauté onion in butter in a large skillet until tender. Add bread cubes, tossing lightly. Stir in 1/3 cup water, tarragon, and pepper. Remove from heat. Add remaining 2 tablespoons lime juice, shrimp, and salt; mix well, and set aside.
Remove fish from lime juice; drain and discard juice. Return 3 fillets to baking dish. Spread stuffing mixture evenly over fillets; top with remaining fillets. Pour tomato juice around fish. Cover dish with aluminum foil. Bake at 350° for 30 minutes; uncover and continue baking 10 minutes or until fish flakes easily when tested with a fork.
Drain cooking liquid from fish into a small saucepan, leaving fish and stuffing mixture in baking dish. Combine cornstarch and remaining water, stirring to remove lumps. Stir dissolved cornstarch into reserved cooking liquid. Bring to a boil; reduce heat, and simmer 2 minutes or until thickened. Remove from heat.
Pour sauce over fish; sprinkle with cheese. Broil 4 inches from heating element 3 minutes or until browned. Cut stuffed fish in half, and serve immediately.