Preheat the oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats.
Using a handheld mixer or stand mixer fitted with the paddle attachment, beat the 1/2 cup (115 grams) softened unsalted butter, 1/2 cup (100 grams) packed light brown sugar, and 1/4 cup (50 grams) granulated sugar together for 1 to 2 minutes, until light and fluffy.
Stir in the 1 large egg and 1 teaspoon pure vanilla extract; stop and scrape down the sides of the bowl as needed.
In a large bowl, whisk together the 1 cup (125 grams) all-purpose flour, 2 teaspoons apple pie spice, 1/2 teaspoon baking soda, and 1/4 teaspoon salt until well combined. Stir in the 1¾ cups (175 grams) old-fashioned rolled oats and set aside.
Slowly add the dry ingredients to the wet ingredients (a little at a time) with your mixer on low; mix until just combined. Gently fold in the 1 cup (120 grams) peeled and finely chopped apple; stir by hand until fully incorporated into the dough. Place dough in the fridge for at least 30 minutes to avoid the cookies spreading too much.
Using a 1.5 tablespoon cookie scoop, scoop balls of the cookie dough onto the prepared baking sheets, and place 3 inches apart on the cookie sheet.
Bake for 12 to 15 minutes or until the tops of the cookies are set. Remove from the oven when cookies still look a bit shiny on the top. They will continue to cook on the baking sheet and set nicely once they have cooled.
Allow to cool on the baking sheet for 5 to 10 minutes, then transfer the cookies to a wire rack to cool completely before icing.
Make the icing: Whisk the 1/2 cup (60 grams) powdered sugar, 2-4 teaspoons milk and a dash of cinnamon together until no lumps remain. If the glaze is too thick, add more milk, and if the glaze is too thin, add more powdered sugar until it has reached your desired consistency. Drizzle a small amount over cookies.