Place butter, sugar, and diced apples into a medium frying pan. Cook over med-high heat, stirring occasionally, until tender and golden. Spread out on a paper towel to cool.
Cookie Dough:
In a medium bowl, whisk flour, cinnamon, baking soda, and salt. Set aside.
In the bowl of an electric mixer, beat butter and sugars on med-high until pale and fluffy (approx. 3 minutes).
Add eggs one at a time, fully incorporating after each addition, and beat on high for 1 minute. Add vanilla.
Turn mixer to low and add flour mixture, mix until combined. Add oats.
Fold in caramelized apples and mix until just combined.
Chill dough in the fridge for at least an hour.
Preheat oven to 350F and line a baking sheet with parchment or a Silpat.
Using a medium cookie scoop, place cookies on baking sheet 2" apart. I did 6 cookies per sheet.
Bake for 12-15 mins or until edges are browned but center is still soft and unset.
Cool for 5 minutes on baking sheet, then transfer to cooling rack to cool completely.
Glaze:
Combine sugar and water in a bowl or measuring cup. Drizzle over cooled cookies with a spoon.