Wash the dirt out of the leek rounds by swishing them around in a big bowl of water.
Place the washed leeks in a big sauté pan set over medium high heat. Season with a big pinch or two of salt, and cook, stirring frequently for 5 minutes. Add the butter and white wine to the pan, reduce the heat to medium low, and cover the leeks partially with a parchment or foil lid. Cook for 30 minutes until the leeks are very soft.
Cut the brioche into cubes and spread on a baking sheet. Toast them for 15 minutes in the oven until they are golden brown.
In a very large bowl, whisk the eggs, then add the thyme, milk, heavy cream, and nutmeg. Add in a big pinch of salt and pepper.
Add the bread cubes, leeks, and cheese to the custard, and toss to combine. Let this soak for about 10 minutes, tossing the bread every few minutes to give it a chance to soak up the custard.
Pour the bowl contents into a 9×13 dish or little ramekins and bake for 70-90 minutes until the top is golden brown and the liquid has been absorbed by the bread (70 for the individual dishes, 90 for the big 9×13). Enjoy!