Preheat oven to 375F (190C) and lightly grease a 9x13 inch baking dish.
Blanch green beans in boiling salted water for 5 minutes, then immediately transfer to an ice bath; drain and set aside.
Heat olive oil in a large skillet over medium heat; sauté mushrooms until browned, approximately 8 minutes.
Stir in garlic and thyme, cooking for 1 minute until fragrant.
Sprinkle flour over mushrooms and cook for 1-2 minutes to remove raw flour taste.
Gradually whisk in vegetable broth and milk; simmer until the sauce thickens, about 5-7 minutes.
Season sauce with salt and pepper, then stir in the blanched green beans.
Transfer the mixture to the prepared baking dish.
In a small bowl, combine panko breadcrumbs, Parmesan, and crispy onions; sprinkle evenly over the casserole.
Bake for 20-25 minutes until the sauce is bubbly and the topping is golden brown.