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bring a large pot of water to boil over high heat.
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add salt and then add the pasta. cook according to the package directions.
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meanwhile, heat the oil in a large skillet over very low heat.
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add the garlic and saute just until it begins to sizzle (do not let it brown).
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stir the garlic and continue to cook, 1 to 2 minutes.
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add the paprika and season with salt and pepper; stir to arrest the browning.
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after 6 minutes, add the broccoli to the pot with the pasta.
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once the pasta is al dente, reserve 1 cup of the cooking water, then drain the pasta and broccoli in a colander.
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add the pasta and broccoli back to the pot and stir in the garlic oil.
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toss together until combined. add the butter and stir to melt.
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add the cheese and stir again.
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moisten the pasta, if necessary, with the reserved pasta water.
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season to taste with salt and pepper.
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garnish with cheese and serve immediately.