Pour olive oil into a 12-inch frying pan or 14-inch wok over medium-high heat; when hot, add mushrooms and garlic and stir until mushrooms begin to brown and liquid has evaporated, 3 to 5 minutes. Add vinegar and thyme; cook, stirring often, to blend flavors, 1 to 2 minutes longer. Add salt and pepper to taste.
In a bowl, whisk eggs, milk, 1/4 teaspoon salt, and 1/8 teaspoon pepper to blend.
Cover bottom of a 3-quart baking dish (such as 9 by 13 in., at least 2 in. deep) with bread slices, trimming if necessary to fit in a single layer. Spread half the mushroom mixture over bread; top with all the crab, half the green onions, and 1 cup cheese. Pour half the custard mixture evenly over cheese. Arrange another layer of bread on top (overlap slightly if desired; save any extra bread for other uses), followed by remaining mushroom mixture, green onions, and 1 cup cheese. Spoon remaining custard mixture evenly over the top. Let strata stand at room temperature about 15 minutes, or cover and chill.
Bake, uncovered, in a 350° regular or convection oven until center is set (cut to test) and top is well browned, 45 to 50 minutes. Cut into rectangles or scoop out with a large spoon to serve.
Cleaning mushrooms: Trim off tough or discolored bottoms of mushroom stems and any bruised spots or blemishes. (For shiitakes and oysters, remove the entire fibrous stem.) For firm mushrooms such as portabellas, wipe dirt off with a damp cloth or place in a colander, rinse thoroughly under cool running water, and pat dry with towels. For delicate mushrooms that have lots of places for dirt to hide, such as chanterelles and hedgehogs, submerge in a bowl of cool water and gently agitate with you hands to loose any particles. Drain, rinse carefully under running water, and gently pat dry with a towel.