Preheat the oven to 475°F (246°C). Line a large baking sheet with parchment paper or a silicone baking mat. Set aside.
Thinly slice the onion, separating the slices. In separate bowls, place flour, Panko + salt + pepper, and whisked egg + milk. Coat onion slices by dipping them in flour, then egg mixture, and finally Panko. Place on the baking sheet. Bake onions until golden brown, approximately 25 minutes, flipping them twice. Set aside.
Reduce oven temperature to 400°F (204°C).
Bring a gallon of water and 1 Tablespoon of salt to a boil in a large saucepan. Add green beans and blanch for 5 minutes. Drain and place in a bowl of ice water to stop cooking. Drain again and set aside.
In a large ovenproof skillet over medium-high heat, melt butter. Add mushrooms, 1 teaspoon salt, and pepper. Cook until mushrooms release moisture (about 5 minutes). Add garlic, stir, and cook for another 2 minutes. Sprinkle flour on top and stir until combined. Add chicken broth, simmer for 3 minutes, then decrease heat to medium-low and add half-and-half. Cook until the mixture thickens (about 10 minutes or more for a thicker sauce).
Remove from heat and add 1/4 of the onions and all of the green beans. Stir to combine. Top with remaining onions and bake until bubbly, about 10-15 minutes.
Remove from the oven and serve warm!