Preheat oven to 450°F (232°C). Line a baking sheet with parchment paper.,Bread onions by dipping in flour, egg-milk mixture, then panko with salt and pepper. Bake for 20 minutes, flipping twice.,Reduce oven temperature to 400°F (204°C).,Blanch green beans in salted boiling water for 5 minutes, then transfer to ice bath. Drain and set aside.,In a large ovenproof skillet, melt butter over medium-high heat. Add mushrooms, 1 tsp salt, and pepper; cook 5 minutes.,Add garlic; cook 2 minutes. Sprinkle in flour and stir. Add broth, simmer 3 minutes. Add half-and-half and cook until thick, about 10 minutes.,Remove from heat. Stir in green beans and a quarter of the baked onions. If not using an oven-safe skillet, transfer to a casserole dish.,Top with remaining onions. Bake for 15 minutes until bubbling.