How to Make It
Preheat the oven to 350°. Spread the hazelnuts in a pie plate and toast for 10 minutes. Let cool, then transfer to a kitchen towel and rub them together to remove the skins. Transfer the hazelnuts to a work surface; coarsely chop.
In a medium skillet, cook the shallots and garlic in 2 tablespoons of the butter over moderately low heat until softened, 4 minutes. Add the mushrooms and thyme; season with salt and pepper. Cook, stirring occasionally, until the mushrooms are tender, 5 minutes. Discard the garlic and thyme. Reserve 1/3 of the mixture. In a food processor, puree the remaining mushroom mixture. Season with salt and pepper.
Cut each skate wing in half crosswise and season with salt and pepper. Spread 4 of the skate pieces with 1/2 tablespoon of the butter and top with one-quarter of the mushroom puree. Cover with the remaining skate wing, pressing the packets together.
In a large skillet, heat the oil until shimmering. Add the skate packets and cook over high heat until browned and crisp on the bottom, 4 minutes. Flip the packets, lower the heat to moderate and add the remaining 2 tablespoons of butter, the reserved mushroom mixture, the hazelnuts, diced lemon and lemon juice. Cook until the skate is just white throughout, 4 minutes. Transfer the skate to plates, and top with the mushrooms, nuts and lemon and serve.