Preheat the oven to 400 degrees F.
In a medium skillet, melt 1 tablespoon butter. When the butter starts to froth and brown slightly on the edges of the pan, shut off the heat and add the leeks. Turn the heat back to medium and season the leeks with salt and pepper. Stir to blend. Cook for 2 to 3 minutes until the leeks become only slightly tender. Transfer the leeks to a baking sheet and place the sheet in the refrigerator.
Crack the eggs into a medium bowl. Whisk the eggs to break them up. Add the slices of goat cheese and season with salt and pepper. Use a whisk to break up the goat cheese and blend it with the eggs. When it is fairly smooth, whisk in the cream, milk, hot sauce, and Worcestershire.
Grease the bottom and sides of the baking dish with the remaining 1 tablespoon butter and fill with the quiche batter.
Place the dish in the center of the oven and bake for about 20 minutes. Open the oven door and gently layer the Brie slices on top of the quiche. Close the oven and bake for an additional 10 to 15 minutes until the top is slightly brown. Allow to cool for 20 to 30 minutes before cutting.
The test: It's a lot harder to check the seasoning of an uncooked quiche batter. If desired, heat a pan, add a touch of butter and cook a tiny bit of the batter for 1 minute. Taste and adjust the seasoning, if needed.