Heat the butter in a large skillet over medium heat. Add the onion and a pinch of salt; cook until the onions are softened, 3-5 minutes. Stir in the mushrooms and garlic; cook until the mushrooms release their liquid, the liquid cooks off, and the mushrooms are nicely browned, about 15 minutes.

Pour in the white wine, increase the heat to high and deglaze. Cook until the wine has evaporated, about 2 minutes. Remove from heat. Stir in the thyme and season with salt and pepper to taste.

Place the bread in a greased 9 x 13-inch baking dish. Layer the mushrooms on top, then the cheeses.

Whisk together the eggs, milk, nutmeg, salt and pepper in a medium bowl. Pour the mixture evenly over the strata.

Cover the casserole dish with plastic wrap and refrigerate overnight.

In the morning, preheat the oven to 350ºF. Take the strata out of the refrigerator and let it sit at room temperature for 30 minutes.

Remove the plastic wrap from the baking dish. Bake the strata for 30-40 minutes, or until the custard is set and the top is browned and crisp.