Preheat the oven to 450°F (232°C). Line a large baking sheet with parchment paper or a silicone baking mat.

Thinly slice the onion and separate the slices.

Prepare three bowls: pour flour in one, mix panko, salt, and pepper in another, and whisk the egg and milk in the last.

Use one hand for dry ingredients and the other for wet. Dip the onion slices in flour, then egg mixture, and finally coat them well with panko.

Place them on the baking sheet and bake until golden brown, about 20 minutes, flipping them twice during the cooking time. Set aside.

Reduce the oven temperature to 400°F (204°C).

Bring a large saucepan with a gallon of water and 1 tablespoon of salt to a boil.

Add the green beans and boil for 5 minutes.

Prepare a bowl of ice water and drain the beans, transferring them to the ice water to stop cooking. Drain again and set aside.

In a large 10- to 12-inch ovenproof skillet over medium-high heat, melt the butter.

Add mushrooms, 1 teaspoon salt, and pepper. Cook for about 5 minutes until mushrooms release moisture.

Add garlic and cook for another 2 minutes.

Sprinkle flour on top and stir until combined.

Add chicken broth and simmer for 3 minutes.

Reduce heat to medium-low and add half-and-half. Cook until the mixture thickens, about 10 minutes.

Remove from heat and add about a quarter of the baked onions and all green beans. Stir well to mix.

Transfer to a greased casserole dish if not using an oven-safe skillet.

Top with the remaining onions and bake until bubbling around the edges, about 15 minutes.

Remove from the oven and serve warm.