Put the butter in a large skillet over medium-high heat. When it melts, add the apples and a pinch of salt, and cook, stirring occasionally, until tender and lightly browned, about 10 minutes.
Meanwhile, put the oats, milk, and cinnamon in a medium pot with a pinch of salt and 1½ cups water. Bring to a boil over medium-high heat, then reduce the heat to low and cook, stirring occasionally, until the oats are tender and the mixture is creamy and thick, 5 to 7 minutes. Turn off the heat, and stir in the vanilla.
When the apples are tender, add the 2 tablespoons brown sugar and stir just until it melts, about 1 minute. Serve the oatmeal topped with the apples and additional brown sugar, if you like.
