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1. Peel the potatoes, cut in half lengthwise, and then thinly slice; place in a large saucepan with cold water to cover by 1 inch. Bring to a boil over high heat, reduce the heat, and simmer until the potatoes are tender, about 10 minutes.
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2. Right after you put the potatoes on to cook, begin the filling: Cook the ground beef in a 12-inch skillet over medium-high heat, breaking up the meat with a wooden spoon and turning frequently until the meat is nicely browned, about 7 minutes. Transfer the beef to a plate. Preheat the oven to 500u00b0F.
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3. Add 2 tablespoons oil to the same skillet. Add the onions and garlic and cook over medium heat, stirring often, until golden and tender, about 6 minutes. Return the browned meat to the pan and stir in the tomato puree and cayenne. Bring to a simmer and cook, stirring occasionally, until the mixture is thickened, about 10 minutes. Stir in the peas and cover to keep warm.
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4. Drain the cooked potatoes well and return them to the pot. Mash well with a potato masher, and then stir in the half-and-half, salt, and pepper to taste. Spoon the warm meat mixture into a 2 1/2-to 3-quart baking dish and sprinkle the Cheddar on top. Spoon the potatoes on top of the cheese, spreading them over the meat, leaving a scant inch border around the edge.
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5. Drizzle the remaining tablespoon of olive oil over the top and then bake until the potatoes are golden, about 10 minutes. Let cool for 5 minutes before serving.
