1. Heat 1 tablespoon of the butter and 1 tablespoon of the oil in a medium saut� pan over medium heat. Add the onions and cook, stirring occasionally, until golden brown and caramelized, about 40 minutes. Season with salt and pepper.
2. Meanwhile, put the potatoes in a medium pot of salted cold water. Bring to a boil and cook until the potatoes are nearly tender and a knife inserted into the centers still meets a little resistance, 10 to 12 minutes. Drain well.
3. Combine the onions and potatoes in a bowl, add the paprika, and season with salt and pepper. Divide the mixture into 8 portions, forming each into a �-inch-thick rectangle.
4. Heat the remaining 3 tablespoons butter and 1 tablespoon oil on a cast-iron griddle or in a large nonstick saut� pan over medium heat. Add the potato rectangles and cook until the undersides are golden brown, about 10 minutes. Flip them over with a spatula and cook until the bottoms are pale golden brown and just set, about 2 minutes.
5. Serve 2 hash browns per person, garnished with the chives
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