Heat the oil in a wide skillet. Cook the mushrooms and onion for 8 to 10 minutes until browned and dry, then add the spinach and cook until wilted. Cool completely.
Butter a 9-by-13-inch dish. Layer in the bread, cooled vegetables and feta.
Whisk the eggs, milk, salt, pepper and oregano and pour evenly over the bread. Press gently so the top cubes touch the custard.
Cover and refrigerate for 8 to 16 hours. Let the dish stand at room temperature for 30 minutes while the oven heats to 350°F.
Bake uncovered for 50 to 55 minutes until golden and set at 160°F in the center. Rest for 10 minutes before cutting.