Heat oil in a medium saucepan over medium-high heat. Add 1/4 cup leek, mushrooms, and garlic; sauté 3 minutes. Add water, rice, 1/8 teaspoon salt, and 1/8 teaspoon pepper; bring to a boil. Cover, reduce heat, and simmer 50 minutes or until rice is tender. Drain rice, reserving 1 cup cooking liquid; set rice aside. Return cooking liquid to pan. Bring to a boil, and cook 8 minutes or until reduced to 1/2 cup. Remove from heat; set broth aside, and keep warm.
Place 1 thyme sprig in the center of each of 4 (10-inch) squares of aluminum foil. Place 1 fish on top of each thyme sprig. Sprinkle remaining salt and pepper over inside cavities of fish. Stuff 1/2 cup rice mixture into cavity of each fish. Wrap fish in foil, twisting ends to seal.
Place foil-wrapped fish on grill rack; grill 10 minutes on each side or until fish flakes easily when tested with a fork. Unwrap fish; remove and discard skin. Set fish aside, and keep warm.
Melt butter in a small saucepan over medium-high heat. Add remaining leek, and sauté 2 minutes or until tender. Divide leek evenly among 4 individual plates, and top with fish. Drizzle 2 tablespoons broth over each fish, and sprinkle with parsley. Garnish with thyme sprigs, if desired.