Melt 2 tablespoons of unsalted butter over medium heat. Add 2 cups of diced Granny Smith apples and 1/2 cup of light brown sugar, cooking for about 6-8 minutes until tender.

Remove from heat and let cool on a paper towel.

In a large bowl, whisk together 1 ½ cups of all-purpose flour, 1 teaspoon of baking soda, 1 teaspoon of ground cinnamon, and 1/4 teaspoon of salt. Set aside.

In a mixing bowl, beat 1/2 cup of unsalted butter with 1/2 cup of light brown sugar and 1/4 cup of granulated sugar until light and fluffy.

Add 1 large egg and 1 teaspoon of vanilla extract, mixing well. Gradually incorporate dry ingredients and fold in 1 1/2 cups of rolled oats and cooled caramelized apples.

Cover the bowl with plastic wrap and refrigerate for at least 1 hour.

Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.

Use a medium cookie scoop to drop dough balls onto the lined baking sheets, spacing them about 2 inches apart.

Bake for 12-15 minutes until edges are golden brown and centers are set but soft. Cool on sheets for 5 minutes before transferring to racks.

Mix 1 cup of powdered sugar with 2 tablespoons of water until smooth. Drizzle over cooled cookies.