Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a bowl, whisk together flour, baking soda, cinnamon, and salt. Set this dry mixture aside.
Using a mixer or a large bowl and a wooden spoon, cream the butter and brown sugar until smooth. Add the egg and vanilla extract, mixing until fully combined.
Gradually add the dry ingredients to the wet mixture, stirring until just incorporated.
Fold in the oats and the apples until evenly distributed.
Scoop tablespoon-sized portions of dough onto the prepared baking sheet, leaving about 2 inches of space between each cookie.
Bake for 10–12 minutes or until the edges are lightly golden. Remove from the oven and allow them to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
While the cookies cool, make the glaze. Whisk together powdered sugar, milk (starting with 1 teaspoon, adding more if needed for consistency), and cinnamon until smooth. Drizzle over the cooled cookies and allow the glaze to set before serving. Enjoy!