To make the apples, add the butter, brown sugar, cinnamon, nutmeg, and diced apples to a saucepan over medium heat. Cook, stirring occasionally, until the apples are tender and golden. Add the cornstarch flurry over the apples. Stir well, and return to low heat for 1–2 minutes, until the mixture thickens. Let cool completely.
In a medium bowl, whisk together the flour, baking soda, cinnamon, and salt.
In a separate bowl, beat the softened butter and sugars for 3–5 minutes, until light and fluffy. Add the egg and vanilla and mix until combined.
Add the dry ingredients and mix until just combined. Fold in the oats, then gently fold in the apples.
Preheat the oven to 350°F and line a baking sheet with parchment paper.
Scoop the dough into portions and place 2 inches apart on the baking sheet. Bake for 15–18 minutes, or until the edges are golden brown and the centers look soft but set.
Let the cookies cool completely before adding glaze.
For the glaze, whisk together the powdered sugar and milk, adding the milk gradually until you reach your desired consistency. Drizzle over the cooled cookies.
