Heat the oven to 450°F (232°C). Cover a large baking sheet with parchment paper or a silicone baking mat and set it aside.
Prepare the onions: Thinly slice the onion and separate the individual pieces. Place the flour in a small bowl. In a second bowl, combine the panko, salt, and pepper. In a third bowl, whisk together the egg and milk. Working with a few onion slices at a time, coat them first in flour, then in the egg mixture, and finally in the seasoned panko. Arrange the coated onions on the prepared baking sheet. Continue until all of the onion has been breaded. Bake for about 20 minutes, turning the onions twice during baking, until golden brown. Set aside. You can blanch the green beans while the onions bake.
Lower the oven temperature to 400°F (204°C).
Blanch the green beans: Bring 1 gallon of water and 1 Tablespoon of salt to a boil in a large saucepan. Add the green beans and boil for 5 minutes. Meanwhile, prepare a large bowl of ice water. Drain the beans and immediately place them in the ice water to halt the cooking process. Drain once more and set aside.
Prepare the casserole sauce: Melt the butter in a large 10- to 12-inch ovenproof skillet over medium-high heat. Add the mushrooms, remaining 1 teaspoon salt, and black pepper. Cook for about 5 minutes, stirring from time to time, until the mushrooms start releasing moisture. Stir in the garlic and cook for another 2 minutes. Sprinkle the flour over the mushrooms and stir until incorporated and the released moisture has been absorbed. Pour in the chicken or vegetable broth and simmer for 3 minutes. Reduce the heat to medium-low and stir in the half-and-half. Continue cooking, stirring occasionally, for about 10 minutes, or longer if you want the sauce thicker.
Take the skillet off the heat. Stir in about one-quarter of the baked onions along with all of the green beans, making sure the beans are evenly coated with the sauce. If your skillet cannot go in the oven, transfer the mixture to a greased 2–3-quart casserole dish. Scatter the remaining onions over the top. Bake for about 15 minutes, until the casserole is bubbling around the edges. Serve warm.
Refrigerate leftovers in an airtight container for up to 4 days and reheat as desired.