Begin by heating vegetable oil in a deep skillet over medium heat until it reaches about 325°F. Toss sliced shallots in a mixture of flour and sea salt. Once the oil is hot, add the shallots, frying until golden brown, about 5-7 minutes. Remove with a slotted spoon and let drain on paper towels.

In a saucepan, heat half-and-half over medium heat until it simmers gently. Stir in the dried shiitake mushrooms and cover, allowing flavors to steep for 30 minutes.

Bring a large pot of salted water to a rolling boil and add fresh green beans. Blanch for 3 minutes until bright green and tender-crisp, then plunge into an ice bath. Drain thoroughly.

After the mushroom mixture has steeped, blend until smooth. In a large saucepan, melt butter over medium heat, stir in flour to create a roux, cooking for about 2 minutes. Whisk in dry sherry, then gradually pour in the blended mushroom cream, whisking until the sauce thickens, around 4-5 minutes. Stir in Gruyere cheese, black pepper, and nutmeg.

Preheat oven to 350°F. In a greased baking dish, layer half the blanched green beans, drizzle with half the creamy sauce, and repeat. Sprinkle extra Gruyere cheese on top.

Bake the casserole for 35-45 minutes, or until the top is golden and bubbly. In the last 5-10 minutes, sprinkle the fried shallots on top.