Preheat oven to 350 degrees.
Coat a 9 x 11 baking dish and a large baking sheet with olive oil or non-stick spray.
Mix together the egg and 2 Tbl water in a large bowl. Add onions to the bowl and stir to coat with the egg mixture. Set aside.
Mix together the panko, parmesan cheese and salt and pepper in another bowl.
Combine the panko mixture with the egg/onion mixture, and spread it out on the baking sheet. Set aside
Heat the 2 Tbl olive oil over medium high in a large saute pan for about 2 minutes, add mushrooms and saute until lightly browned. Transfer cooked mushrooms to a separate bowl.
Make a white sauce by melting the 3 Tbl butter over high heat, reduce heat to medium and add the flour and stir until thickened, about 1-2 minutes. Add the milk and vegetable broth slowly, while whisking continuously. Keep whisking over medium heat until the whole sauce is about the consistency of pancake batter. Add the cooked mushrooms to the white sauce.
Place the uncooked green beans in the baking dish, pour the white sauce/mushroom mixture over the green beans. Sprinkle the dried herbs over the top, if desired. Bake the green bean/sauce combo AND the onion breadcrumb mixture at 350 degrees for about 20 minutes. Stir the onion mixture around a bit on the baking sheet, rotate the green bean dish around in the oven. Bake both for an additional 20 minutes.
Remove the green bean casserole from the oven and turn the temp up to 425. Bake the onions for an additional 10-15 minutes or until crispy golden brown. Remove from oven and use tongs or a spatula to gently place the onions on top of the green bean mixture. It's piping hot, so allow to cool for a bit before serving.