Preheat the oven to 350 degrees F. Brush a 9-inch pie plate with the butter and sprinkle the Parmesan evenly on top. Put on a rimmed baking sheet.
Whisk the half-and-half, eggs, egg yolks, cayenne, 1/2 teaspoon salt and a few grinds of pepper in large bowl. Spread the broccoli in the prepared pie plate and top with the goat cheese. Pour the egg mixture over top and sprinkle with parsley.
Bake until the quiche is just set in the center, 40 to 50 minutes. Transfer to a wire rack to cool and then cut into wedges.
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