Preheat the oven to 375ºF. In a large bowl, whisk the eggs and milk together with a generous pinch of salt and pepper. Set aside.
Heat the oil in a large skillet over medium heat. Tilt the pan so the oil coats the bottom and sides of the pan. Add the onion and cook, stirring occasionally, for 2 minutes. Add the asparagus and mushrooms. Cook, stirring occasionally, for 2 minutes. Season with a pinch of salt and pepper. Add the garlic and stir for 1 minute. Pour the egg mixture over the vegetables. Allow to cook, undisturbed, for 1-2 minutes.
Transfer to the oven and bake for 10-15 minutes or until set. Sprinkle with sliced basil, cut into wedges, and serve.