Preheat the oven to 450 degrees . In a large skillet, heat the olive oil over medium-high heat. Add the onion and cook, stirring often, until softened, about 5 minutes. Add the mushrooms and cook, stirring occasionally, until theyve released and absorbed most of their juices, about 5 minutes.
In a large bowl, combine the tuna and mayonnaise, breaking up the tuna. Stir in the onion mushroom mixture, then stir in two-thirds of the cheese.
In a small bowl, whisk together the milk, eggs, flour and butter; stir into the tuna mixture. Transfer to a 10-inch deep-dish pie pan and bake until browned, about 35 minutes. Top with the remaining cheese and broil until the cheese has melted.