Preheat the oven to 375 degrees F (190 degrees C). Spray a 9- by 13-inch (33x23cm) baking pan with nonstick cooking spray.
Heat 1 tablespoon of the oil in a skillet over medium-high heat. Add the mushrooms then cook, stirring occasionally, until they release their juices and start to brown, about 5 minutes. Add the spinach and cook until wilted, about 2 minutes. Season to taste with salt and pepper. Remove from the heat and set aside to cool slightly.
In a large bowl, whisk together the eggs and milk. Season with salt and pepper. Stir in the cooled spinach mixture, then transfer to the prepared baking pan.
Bake the omelette until puffed and set, 15 to 20 minutes. Remove from the oven and let cool 10 minutes before serving.