In a medium saucepan, melt the butter over medium heat. Add the apple, 2 tablespoons of brown sugar, and cinnamon. Cook, stirring frequently until the apples are softened and fragrant, 2 to 3 minutes. Remove a quarter of the apples to a small bowl.
In the saucepan with the remaining apples, add the oats, salt, the remaining 1 tablespoon of brown sugar, and 2 cups of water. Bring the mixture to a simmer and cook, stirring occasionally, until the oats are softened, 3 to 5 minutes more.
To serve: divide the oatmeal between two bowls and top each with the heavy cream. Top with the reserved apples and sprinkle with brown sugar, cinnamon, and walnuts, if you like.
