Heat vegetable oil in a deep skillet to 325°F. Combine flour with a pinch of salt and toss the sliced shallots to coat. Fry shallots until golden brown, about 4-5 minutes, then drain on paper towels.

In a small saucepan, heat half and half over medium heat until simmering. Add dried shiitake mushrooms and cover, steeping for 30 minutes.

Trim the ends of the green beans. Boil a large pot of water and blanch green beans for 3 minutes, then transfer to an ice water bath.

Purée the mushroom cream until smooth. In another saucepan, melt butter, whisk in flour to make a roux. Gradually add puréed mushroom cream, followed by dry sherry. Season with salt, pepper, and nutmeg, then fold in Gruyere cheese until melted.

In a greased baking dish, layer half of the blanched green beans and drizzle with half of the mushroom sauce. Repeat with remaining beans and sauce. Top with remaining Gruyere cheese.

Preheat oven to 350°F. Bake uncovered for 35–45 minutes or until golden and bubbling. If the top browns too quickly, tent with foil.