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Preheat the oven to 400 degrees Fahrenheit. Rub the chicken thighs with the 2 teaspoons of olive oil and season with the salt and pepper. Place on a lipped baking sheet, skin side up, and roast in the oven until the skin is golden, the juices run clear, and a thermometer inserted into the center of the thigh reads 165 degrees Fahrenheit, about 30 to 40 minutes. Remove the pan from the oven and allow the chicken to cool until cool enough to handle.
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Save the juices that have collected on the bottom of the lipped baking sheet and set them aside for now. Discard the chicken skin, remove the meat from the bones, and chop the meat into roughly 1-inch cubes. Discard the chicken bones and set the meat aside.
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Bring a large pot of water to a boil, then season generously with salt.
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While the water is coming to a boil, you can prepare the sauce. Heat the 2 tablespoons of olive oil in a large Dutch oven over medium low heat. Add the garlic and sauté until softened and very lightly golden around the edges, about 5 minutes, stirring every minute.
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The following steps should happen very quickly, so pay close attention and read ahead to be prepared. Add the orecchiette to the pot of boiling water and prepare al dente according to the package directions. When you eventually strain the pasta, make sure to reserve 3/4 cup of the water the pasta was cooked in.
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While the orecchiette is cooking, add the reserved chicken pan juices and the lemon juice to the pot of garlic and whisk to combine.
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Add the rosemary and lemon zest and crumble the chèvre over the sauce, stirring until the chèvre is melted. The sauce will look like it’s separated, but it will come together in a bit.
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Take 3/4 cup of water from the boiling pot of pasta using a heat-proof container like a glass measuring cup. Slowly add it to the pot of sauce in a thin steady drizzle, whisking constantly, until the sauce becomes creamy.
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Add the strained orecchiette, the chopped chicken, peas, chives, and black pepper to the pot of sauce and toss gently to combine and coat the pasta in the sauce. Remove from heat, salt to taste, if desired, and serve immediately.