Heat butter in a small frying pan over medium heat until foam subsides. Add mushrooms and cook, stirring occasionally, until they give up most of their liquid and are tender and beginning to brown. Turn out into a small bowl and set aside.
Whisk the eggs and milk lightly together in a small 1-quart saucepan. Add the truffle oil and about 4 pinches of salt. Over low heat, stir the eggs constantly with a wooden spatula, keeping contact with the bottom of the pan. When eggs begin to come together, add mushrooms. In 5 minutes or so, the eggs will be creamy, soft, and ready to serve. Best eaten with a spoon, with brioche toasts alongside (or pile the eggs on the toast).