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In a large skillet, combine the broccoli, butter, olive oil, garlic, and salt to taste.
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Place the skillet on the stove and cook on medium-low heat for 4 minutes, stirring frequently. Add a few tablespoons of water, as needed, to ensure the broccoli does not burn.
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Add the cream and the peas to the skillet and cook for 3 minutes.
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Turn off heat and mix in the cheese and the dill or mint.
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Cook pasta according to package instructions.
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Drain the pasta.
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Add the pasta to the skillet and toss with the sauce.
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Serve with grated Pecorino Romano cheese.