Lightly butter a large baking dish (I used a round 10-inch casserole dish) and set it aside.
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, stir 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 meantime, mix eggs, cheese, parsley, and the stock in a mixing bowl.
Place the now-cooled cooked mushroom and vegetable in a large bowl and add in the brioche cubes. Pour in the egg and milk mixture. Gently stir to combine. Transfer it to the prepared buttered baking dish. Using a spatula distribute it equally in the casserole dish.
Cover it with stretch film and place it in the refrigerator. Let it rest for half an hour (up to 3 hours) for all the juices to be absorbed.
An 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 chopped parsley. Serve while it is still warm.