Add enough oil to a large, deep skillet until it reaches 1 ½ inches up the sides. Turn heat to medium-high.

Meanwhile, lightly whisk the eggs in a bowl; set aside. Whisk flour and salt in a second bowl; set aside. Dip onions in the eggs and then in the flour mixture. Transfer to another bowl until oil is hot. Add onions to oil and fry until golden brown; transfer to a paper towel-lined plate. I fried my onions in two batches.

Preheat the oven to 350 degrees F.

Bring a large pot of water to boil and add green beans. Cook for 5 minutes, drain and immediately place in a large bowl of ice water to stop the beans from further cooking. (In lieu of an ice water bath, I sometimes run cold water over the beans until they are cool to touch.) Drain and set aside.

Melt two tablespoons of butter in a large pan over medium-high heat. Add half of the mushrooms (or less), making sure that there is only one layer of mushrooms so they can evenly brown — don’t crowd them (see my instructional on pan-frying mushrooms here). Remove first batch of browned mushrooms and set aside. Cook the second batch of mushrooms until browned on both sides, adding more butter as needed if the pan is dry. Add ½ tablespoon of butter to the pan followed by the nutmeg and the garlic, making sure the garlic cooks in the melted butter. Return the first batch of mushrooms to the pan and sprinkle the flour over the mushrooms, stirring to combine. Cook for 1 minute. Add the stock and simmer for 1 minute. Add the cream and half-and-half and cook until the sauce thickens, approximately 6 to 8 minutes. Add salt and pepper to taste.

Remove from heat and stir in ¼ of the fried onions and all of the green beans. Transfer mixture to a casserole dish and bake until bubbly, approximately 15 minutes. Add the remaining onions to the top of the casserole and cook for an additional 5 minutes. Serve immediately.