How to Make It
Preheat the oven to 275ºF. Slice the brioche into rough 1-inch chunks and lay on a large sheet pan. Bake until dry (depending on how stale the bread was when you started this can take anywhere from about 45 minutes-1 hour). When the breadcrumbs are finished, let them cool slightly, then transfer to a large bowl.
In a large skillet, melt the butter and let it foam over medium heat. Add the onions and sauté until translucent, then add the celery, mushrooms, 1 teaspoon salt and pepper. Cook until the vegetables are soft, about 5 more minutes.
Pour the vegetables into the bowl with the bread, and mix in the herbs and vegetable stock. Taste, and add remaining salt and more pepper if needed.
Increase the oven temperature to 375ºF and butter a 13x9-inch casserole dish. Pour the casserole mixture into the dish. Note: at this point, the casserole can be covered with plastic wrap and chilled overnight in the refrigerator.
Cover the casserole with foil and bake for 35 minutes, until it seems significantly firmer. Uncover, and bake for an additional 15 minutes.
Pull the casserole out of the oven. Use ⅓-cup measuring cup to make 6-8 impressions in the casserole. Crack an egg into a small dish, then pour it into one of the impressions. Repeat until each hole is filled with an egg.
Return the casserole to the oven and bake for 15-25 minutes, or until the egg whites have set and the yolks are cooked to your liking.