Place an oven rack in the center position and preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Wash, peel, and core the apples, then dice into very small 1/4-inch pieces. Measure 1 1/2 cups and pat dry with a paper towel if especially juicy.
In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg for about 30 seconds. Set aside.
In a large bowl, beat the softened butter, brown sugar, and granulated sugar on medium speed for 2 to 3 minutes until pale, fluffy, and creamy, scraping down the sides as needed.
Add the eggs one at a time, mixing about 30 seconds after each, then pour in the vanilla and mix until combined. Scrape the bowl again.
Add the flour mixture in two or three additions, mixing on low or stirring just until no dry streaks of flour remain. Do not overmix.
Fold in the rolled oats until evenly coated, then gently fold in the diced apples until spread throughout the thick, chunky dough.
Use a medium cookie scoop or heaping tablespoon to portion dough balls onto the lined sheets about 2 inches apart. Gently press the top of each ball down slightly.
Bake one sheet at a time on the center rack for 11 to 13 minutes, until the edges are golden brown and the centers look set but still soft. Do not wait for the tops to brown.
Let the cookies rest on the baking sheet for 5 minutes, then transfer to a wire rack and cool for at least 20 to 30 minutes until completely cool to the touch.
In a small bowl, whisk the powdered sugar, cinnamon, vanilla, salt, and milk until completely smooth. The glaze should fall from the whisk in a thick ribbon; adjust with more milk or powdered sugar as needed.
Set parchment under the cooling rack, then drizzle or dip the cooled cookies with glaze. Let sit for 20 to 30 minutes until the glaze is dry to the touch and fully hardened.