Preheat the oven to 350 degrees F. Brush a 9-inch pie plate with the butter and sprinkle the Parmesan evenly on top. Place the pan on a rimmed baking sheet.
Whisk together the half-and-half, eggs and egg yolks in large bowl. Season with 1/2 teaspoon salt, a few grinds of pepper and the paprika. Put the cheese and salami in a medium bowl and toss to combine; spread the mixture in the prepared pie plate. Pour the egg mixture over the top and sprinkle with the basil.
Bake until the quiche is just set in the center, 40 to 50 minutes. Transfer to a wire rack to cool. Scatter more basil over the top and slice into wedges.
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