Fusilli with Garlicky Broccoli

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Kosher salt and freshly ground black pepper

1 pound fusilli pasta 

2 pounds broccoli, stems removed and heads trimmed into florets

2 pounds broccoli, stems removed and heads trimmed into florets 

1/3 cup extra-virgin olive oil 

15 to 20 cloves garlic, finely minced 

1/4 teaspoon paprika 

2 tablespoons unsalted butter 

1/2 cup freshly grated Parmigiano-Reggiano cheese, plus more for garnish