Thaw fish, if frozen. Rinse fish; pat dry with paper towels. Thinly slice one of the lemon halves; cut each slice in half. Set aside.
For marinade, strain the juice from the remaining lemon half into a small bowl. Stir in oil, the snipped or dried thyme, the salt, and crushed red pepper. Remove 2 tablespoons of the marinade. Add the mushrooms to the remaining marinade; toss to coat. Cover and marinate at room temperature until ready to serve.
Place the halved lemon slices and thyme sprigs inside the fish cavities. Cut off the root ends and 1 inch off the tops of the green onions. Brush the skin and inside flesh of fish and the onions with the 2 tablespoons marinade. Place the fish in a greased grill basket.
Grill fish on the rack of an uncovered grill directly over medium coals for 6 minutes. Turn fish; place green onions on grill rack. Grill for 4 to 6 minutes more or until fish flakes easily with a fork and onions are tender, turning onions once halfway through grilling. (Or, broil fish on the greased unheated rack of a broiler pan about 4 inches from the heat for 6 minutes. Turn fish; place green onions on rack. Broil for 4 to 6 minutes more, turning onions once halfway through broiling.)
To serve, cut the green onions into 2 1/2-inch pieces. Serve the fish with the green onions and marinated mushrooms.