Steam eggs in a vegetable steamer 16 minutes; cool slightly, and peel. Slice eggs in half lengthwise, and remove yolks; mash yolks.
Sauté mushrooms, shallots, thyme, salt, and pepper in butter for 4 minutes. Stir mushroom mixture, mayonnaise, and yogurt into yolks. Divide among whites; top with shaved Manchego.