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.
When is it done? Gently tap or shake the pan; if the filling jiggles slightly in the middle but not on the sides, the quiche is set.