Preheat oven to 375 degrees. Brush a 9-by-2 1/2-inch springform pan with butter. Line the sides of the pan with strips of waxed paper (the same height as pan); brush paper with butter.
Squeeze excess moisture from hash browns. Mix in a bowl with butter, 1 egg, 1 teaspoon coarse salt, and 1/4 teaspoon pepper. Pat into bottom and up sides of prepared pan, using a moistened dry measuring cup. Place on a rimmed baking sheet; bake until set, 15 to 20 minutes.
In a large bowl, whisk sour cream, goat cheese, 1 1/2 teaspoons coarse salt, and 1/4 teaspoon pepper until well combined; whisk in 11 remaining eggs. Pour into crust, and sprinkle with scallions. Bake until set, 45 to 50 minutes. Unmold quiche, and peel off waxed paper before serving.