Preheat the oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats.
In a large mixing bowl, whisk together the flour, cinnamon, nutmeg, baking soda, and salt until well combined. Stir in the old-fashioned rolled oats and set aside.
Using a handheld mixer or stand mixer fitted with the paddle attachment, beat the butter, brown sugar, and granulated sugar together for 1 to 2 minutes or until fully combined. Mix in the egg and vanilla extract, making sure to stop and scrape down the sides of the bowl as needed.
Add the dry ingredients to the wet ingredients and mix until just combined. Gently fold in the chopped apple until fully incorporated into the cookie dough.
Using a 1.5 tablespoon cookie scoop, scoop balls of the cookie dough onto the prepared baking sheets, making sure to leave a little room between each one.
Bake for 12 to 15 minutes or until the tops of the cookies are set.
Remove from the oven and allow to cool on the baking sheet for 5 to 10 minutes, then transfer the cookies to a wire rack to cool completely.