Preheat oven to 350°F. Line two baking sheets with parchment paper.
Toss diced apple with lemon juice and cornstarch in a small bowl; set aside.
Cream butter, brown sugar, and granulated sugar in a large bowl until light and fluffy.
Beat in vanilla and milk until smooth and slightly creamy.
Whisk flour, oats, baking powder, baking soda, cinnamon, nutmeg, and salt in a separate bowl.
Add dry ingredients to the butter mixture and mix just until a thick dough forms.
Fold in coated apples and nuts, if using, until evenly distributed.
Scoop dough in heaping tablespoonfuls (about 2 tbsp each) onto sheets, spacing 2 inches apart. Gently flatten.
Bake 11–13 minutes, until edges set and tops are lightly golden but centers look soft.
Cool cookies on the sheet 5 minutes, then transfer to a rack to finish cooling.
Drizzle with caramel, if desired, once cookies are fully cooled.