Preheat the oven to 350°.
Whisk together eggs, milk, cream, salt, pepper and oregano until well combined.
Add cubed bread, cover and let soak at least 30 minutes, or until it has soaked up about ½ of the liquid.
While bread is soaking, prepare your vegetables and grate the cheese. Set aside.
After the bread has soaked, stir in the prepared vegetables and ¾ of the cheese. Pour entire mixture into an oiled 9- by 13-inch casserole dish. Top with remaining cheese. Cover with parchment paper and then cover tightly with foil.
Bake covered for 25 minutes. Remove cover, rotate casserole and bake another 15–20 minutes until the custard in the middle has set and the top is golden brown. Allow to stand at least 10 minutes before serving. Serve warm or at room temperature.