Preheat the oven and prepare a baking sheet with parchment paper.

Slice the onion thinly and separate the rings.

Set up three bowls for coating: one with flour, one with beaten egg and milk, and one with panko mixed with salt and pepper.

Coat each onion slice by dipping in flour, then egg mixture, and finally breadcrumbs. Arrange on the baking sheet and bake until golden and crisp, flipping occasionally, then set aside.

Lower the oven temperature. Bring a large pot of salted water to a boil and cook the green beans briefly until just tender. Immediately transfer them to ice water to stop the cooking process, then drain and set aside.

In a large oven-safe skillet, melt butter over medium-high heat. Add mushrooms, salt, and pepper, cooking until they release their moisture. Stir in garlic and cook until fragrant.

Sprinkle flour over the mixture and stir until absorbed. Pour in the broth and let it simmer, then reduce the heat and add the half-and-half. Cook until the sauce thickens to your desired consistency.

Remove the skillet from heat and stir in the green beans along with a portion of the baked onions. Mix well so the beans are fully coated in the sauce. If not using an oven-safe skillet, transfer the mixture to a greased casserole dish.

Top with the remaining onions and bake until the casserole is bubbling around the edges and heated through. Remove from the oven, let it rest briefly, and serve warm.