Preheat oven to 375°F (190°C). Grease a 9 x 13-in. (3 L) baking dish. In a skillet, heat oil over medium-high heat. Add onion, mushrooms, oregano and 1/2 tsp (2 mL) each salt and pepper and cook, stirring, until all liquid from mushrooms evaporates, about 8 min. Add red pepper and cook until softened, about 5 min. Add spinach and cook until wilted, about 2 min. Set aside.
In a large bowl, stir together bread cubes, mushroom mixture and goat cheese. Spread mixture in prepared baking dish. In a bowl, whisk together eggs, milk and remaining salt and pepper. Pour over bread mixture and let stand, 15 min. or overnight.
Bake on the bottom rack of oven until puffed, golden and set, 40 to 45 min.
Make ahead: Prepare this dish the night before and pop it in the oven to bake while you open presents.