Cook the Apple: Melt the butter in a medium skillet over medium heat. Add the grated apple and cinnamon. Cook for 5–7 minutes, stirring frequently, until the apple is tender and most of the moisture has cooked off. You don't want liquid pooling in the pan. Transfer the apple to a bowl and let cool completely. You should have about 1/2 cup cooked apple.
Brown the Butter. Melt the butter in a medium saucepan over medium heat. Continue cooking, stirring occasionally, until the butter foams and develops golden-brown bits and a nutty aroma. Remove from the heat and pour into a bowl. Let cool to room temperature. You can put it in the refrigerator for 15 minutes to speed up the process.
Combine Dry Ingredients. In a medium bowl, whisk together the oats, flour, baking soda, salt, cinnamon, allspice, and nutmeg.
Combine Wet Ingredients. In a large bowl, whisk the cooled brown butter, brown sugar, and granulated sugar together. Add the egg, egg yolk, and vanilla and whisk until smooth. Stir in the cooled apple.
Make the Dough. Add the dry ingredients and stir just until combined.
Chill. Cover the bowl and chill the dough for at least 30 minutes. You can chill the dough for up to 24 hours.
Form Dough Balls. Preheat oven to 350°F. Line baking sheets with parchment paper or silicone baking mats. Scoop the dough into 1 1/2 tablespoons portions and place about 2 inches apart on the prepared baking sheets. Do not flatten the dough.
Bake. Bake for 10–12 minutes, or until the edges are lightly golden and the centers are still soft. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Make the Salted Caramel Glaze. In a medium bowl, whisk together the confectioners' sugar, salted caramel sauce, milk, and vanilla until smooth. If the glaze is too thick, add a little more milk. If it is too thin, add a little more confectioners' sugar.
Add the Glaze. Drizzle the glaze over the cooled cookies and let the glaze set before serving.