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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.
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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.
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Place the bread in a greased 9 x 13-inch baking dish. Layer the mushrooms on top, then the cheeses.
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Whisk together the eggs, milk, nutmeg, salt and pepper in a medium bowl. Pour the mixture evenly over the strata.
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Cover the casserole dish with plastic wrap and refrigerate overnight.
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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.
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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.