Preheat oven to 400 degrees F. Brush a 9-inch glass or ceramic pie pan with the soft butter and sprinkle half the Parmesan on top. Place the pan on a baking sheet.
Squeeze the spinach in batches, very firmly by hand, to remove excess water. Put spinach in a food processor, along with the eggs, egg yolks, half-and-half, Gouda, flour, nutmeg, and salt and pepper to taste. Pulsing until mixture is an even green color with bits of spinach, about 30 seconds. Add the scallions and pulse briefly to mix. Pour mixture into the prepared pan. Sprinkle the top with the remaining 2 tablespoons Parmesan and paprika.
Place on the lower rack of the oven bake until quiche is just set in the center, about 40 to 45 minutes. Cool completely on a rack before serving.
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