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

For the onions: Thinly slice the onion, and separate the slices. Pour flour in one small bowl, panko + salt + pepper into a medium bowl, and whisk the egg and milk together in another small bowl. Bread the onion slices by dipping them in flour, then egg mixture, and finally coat with panko. Place on baking sheet. Bake onions until golden brown, about 20 minutes.

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

Blanch the green beans: Bring a gallon of water and 1 Tablespoon of salt to a boil in a large saucepan. Add the green beans and boil for 5 minutes. Transfer to a bowl of ice water to stop cooking. Drain and set aside.

For the casserole: Melt the butter in a large ovenproof skillet. Add the mushrooms, 1 teaspoon salt, and pepper. Cook until mushrooms release moisture, about 5 minutes. Add garlic, cook for 2 minutes. Stir in flour until combined. Add broth and simmer for 3 minutes. Stir in half-and-half and cook until thick, about 10 minutes.

Remove from heat and add a quarter of the onions and all of the green beans. Stir to combine. Top with remaining onions and bake until bubbling, about 15 minutes.

Serve warm.

Leftovers keep well in an airtight container in the refrigerator for up to 4 days. Reheat as desired.