Butter a large oven-proof dish. You can use a large oval casserole dish (like the one I used) or a 9X13 Pyrex.
Heat olive oil in a Dutch oven (or any other heavy bottom pan) in medium heat. Add leeks and shallots. Cook, stirring occasionally, for 3 -5 minutes, or until wilted. Stir in the mushrooms, garlic, salt, pepper, and thyme, and cook 8-10 minutes or until most of the juices evaporated.
Stir in the butter. When the butter melts, add in the wine. Turn it down to medium-low heat, and cook for 5-8 minutes or until most juices are gone. When cooked, fish out the thyme sprigs, and allow it to cool for 10-15 minutes, off the heat.
In the mean time, mix eggs, cheese, parsley, and the stock in a mixing bowl.
In a large bowl, fold the mushroom mixture gently in the brioche cubes. Once mixed, pour the milk mixture into bowl. Gently stir to combine. Transfer it to the buttered dish. Using a spatula distribute it equally in the casserole dish.
Cover it with stretch film and place it in the refrigerator. Let it sit in there for half an hour (up to 3 hours) for all the juices to be absorbed.
hour before serving, pre-heat your oven to 350 degrees. Bake for 30 – 35 minutes or until the top of the stuffing is golden.
Sprinkle it with a couple of tablespoons of chopped parsley and cayenne pepper, if preferred.