Position a rack in the center of the oven and preheat to 400 degrees . In a medium skillet, melt the butter over medium-high heat until frothy and brown around the edges of the pan; remove from the heat. Stir in the leeks and return the pan to medium heat; season with salt and pepper. Cook until the leeks are slightly tender but not soft, about 3 minutes. Spread on a baking sheet; refrigerate.
Meanwhile, in a medium bowl, add the eggs, whisking to break them up. Crumble in the goat cheese and whisk until the mixture looks fairly smooth; season lightly with salt and pepper. Whisk in the cream, milk, hot sauce and worcestershire.
Grease an 8-inch square baking dish. Fill it evenly with the chilled leeks, then pour in the egg mixture. Bake for 20 minutes. Remove from the oven and layer the brie pieces on top. Bake again until the quiche is slightly browned and just set, 10 to 15 minutes. let cool for 20 to 30 minutes. Slice into 8 even squares and top with the scallions.