Step 1

Make the casserole: Melt the butter in a large Dutch oven over medium heat. Add the diced onion; cook, stirring, until softened, 7 to 10 minutes. Add the mushrooms; sauté until softened and most of the liquid has evaporated, 10 minutes. Sprinkle in the flour; cook, stirring, 1 minute. Add the broth and half-and-half. Bring to a boil, then reduce the heat to medium low and simmer until thickened, about 5 minutes.

Step 2

Bring a pot of salted water to a boil. Add the green beans and cook until crisp-tender, 7 to 10 minutes. Drain; shake o any excess water and add to the mushroom sauce. Cook, stirring occasionally, until the beans are very tender, 10 to 12 minutes. Season with salt and pepper.

Step 3

Make the topping: Soak the sliced onion in the buttermilk, 5 minutes. Combine the flour and 1 teaspoon salt in a medium bowl. Heat 2 to 3 inches vegetable oil in a large pot until a deep-fry thermometer registers 350 degrees F. Working in batches, drain the onion, toss with the flour and shake o the excess, then fry until golden, 4 to 6 minutes. Drain on paper towels.

Step 4

Transfer the green bean mixture to a 4-quart baking dish and top with the fried onion. (The casserole and topping can be made up to 2 hours ahead; refrigerate the casserole. Bring to room temperature; reheat at 350 degrees F, 30 minutes. Top with the fried onion; bake 5 more minutes.)