Trim off and discard any stems from mushroom caps; gently rinse caps and drain well. Rinse green onions; trim off and discard ends. In a small bowl, mix olive oil and lemon juice.
Brush smooth sides of mushroom caps and the green onions with oil mixture. Lay mushrooms, smooth side down, and green onions on a grill over hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid if using a gas barbecue. Turn onions as needed until lightly browned on all sides, 2 to 3 minutes total. With tongs, transfer to a plate. Cook mushrooms without turning until they are limp and begin to release their juices, 3 to 5 minutes. With a wide spatula, transfer, smooth side down, to a 9- by 13-inch baking pan.
Drizzle mushrooms evenly with vinegar and sprinkle with pepper. Add 1/4 cup vegetable stock to pan. Cover and bake in a 300° regular or convection oven until hot, 5 to 10 minutes.
Meanwhile, in a 1- to 2-quart pan over high heat, bring remaining 1/2 cup stock to a boil. Stir in couscous. Cover and remove from heat. Let stand until liquid is absorbed, about 5 minutes. Fluff couscous with a fork. If it seems too dry, stir in 3 to 5 tablespoons liquid from mushroom caps. Stir in parsley.
Set mushrooms, smooth side down, on plates. Top equally with couscous. Spoon pepper coulis equally around mushrooms and garnish with green onions. Add salt to taste.