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In a shallow dish, combine the flour with 1/2 teaspoon salt and 1/4 teaspoon freshly ground black pepper. Dredge the skate in the seasoned flour, shaking to remove any excess.
In a large skillet, heat the vegetable oil and melt 2 tablespoons of the butter over medium-high heat. Add the skate and cook until golden brown on both sides, about 3 minutes per side. Transfer to a warm platter and cover to keep warm while making the sauce.
With a paper towel, carefully wipe out skillet and return to medium-high heat. Add the remaining 6 tablespoons butter and cook, swirling occasionally, until the butter just begins to brown and has a nutty fragrance. Add the vinegar, parsley, and capers and stir to combine. Remove from the heat and adjust seasoning to taste.
Pour the sauce over the skate fillets and serve with the potatoes. Garnish with parsley and lemon slices, and serve immediately.
In a medium saucepan, cover the potatoes with salted water. Bring to a boil and cook until tender, about 20 minutes. Drain. When cool enough to handle, slice into thin rounds.
In the saucepan, melt the butter over medium heat, and add the salt and pepper. Add the potatoes and parsley, and toss to coat.