Preheat oven to 450 degrees F. Lightly coat a 3-quart rectangular baking dish with nonstick cooking spray. In a very large high-sided skillet, heat oil over medium heat. Add mushrooms, green beans, and onion. Cook, stirring occasionally, for 7 to 8 minutes or until green beans are nearly tender and liquid has evaporated. Add garlic and cook for 1 minute more.
Increase heat to medium-high. Add half-and-half. Boil gently, uncovered, for 5 minutes or until half-and-half has reduced enough to lightly coat the vegetables. Stir in Worcestershire and thyme.
Spoon mixture into prepared baking dish. Top with tuna, cheese, and potatoes. Bake, uncovered, for 30 to 35 minutes or until potatoes are browned and center is bubbly. Let stand 10 minutes before serving. Sprinkle with green onions.