Place half of the bread in a 9x13" baking dish.
Sprinkle 1 1/2 cups of cheese over the bread.
Add the ham, onions, mushrooms, and remaining bread.
Whisk the eggs with the cream, salt, and thyme. Pour this custard over the ingredients in the pan.
Cover and refrigerate overnight (or at least one hour).
Bring the casserole to room temperature before baking. Sprinkle the casserole with the remaining Swiss cheese.
Bake at 325F, uncovered, for 40-50 minutes, until the casserole puffs and the custard is set.
Let rest 10 minutes before serving.