In a medium skillet, melt 1 tablespoon of the butter. Add the shiitakes, season with salt and pepper and cook over high heat, stirring, until tender and golden, 8 minutes. Finely chop the mushrooms.
In a large saucepan, bring the remaining 8 tablespoons of butter, the milk, cayenne and 2 1/2 teaspoons of salt to a boil. Remove from the heat and add the flour, stirring vigorously with a wooden spoon until combined. Cook the dough over moderate heat, stirring constantly, until it comes away from the sides of the pan, about 2 minutes. Remove from the heat. Using a handheld electric mixer, beat in the eggs, one at a time, beating well between additions, until a soft, silky dough forms. Stir in the shiitakes, Gruyère, Parmesan and parsley.
In a large saucepan, heat the oil to 375°. Using a small ice cream scoop or tablespoon and working in batches, add 12 level scoops of the batter to the hot oil. Fry until the beignets are golden brown and cooked through, about 4 minutes. Using a slotted spoon, transfer the beignets to a paper towel-lined rack to drain. Serve hot.
The cheese beignets can be rewarmed in a hot oven until crisp.