Step 1

Generously butter an 11x7-inch (1 1/2-quart) baking dish and set aside.

Step 2

In a large skillet or braiser, heat the olive oil over medium to medium-high heat. Add the shallot and cook until tender, about 3 minutes, followed by the garlic, sautéing for another 30 seconds or so. Add the mushrooms, the leaves from the thyme sprigs, and a pinch of salt. Cook until mushrooms are golden, 8 to 10 minutes. Pour in the wine and increase heat to high. Cook, letting the wine bubbling vigorously, until all the liquid cooks out.

Step 3

Meanwhile, whisk together the eggs, cream, milk, Dijon, salt, and a pinch of black pepper in a large mixing bowl. Arrange half of the bread cubes to the bottom of the prepared baking dish, followed by half the mushrooms and half the cubed brie. Pour half of the egg mixture over the top. Arrange the remaining bread, followed by the mushrooms and brie, over the top. Pour the remaining egg mixture evenly over the bread and push down gently to make sure that every piece gets saturated. Cover and refrigerate for 2 hours or overnight.

Step 4

Preheat the oven to 350°F. Cover the baking dish with aluminum foil and place on a baking sheet. Bake the strata for 60 minutes. Remove aluminum foil, add grated Parmesan and continue baking, uncovered, until puffed and golden, 15 to 25 minutes. Let stand for 10 to 15 minutes before serving.

Step 5

To freeze: Wrap baking dish tightly with plastic wrap, followed by a layer of aluminum foil. Label with date and cooking instructions and freeze for later use. Remove from the freezer and thaw in the refrigerator overnight. Bring to room temperature 30 minutes before cooking; bake according to instructions.