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.