Prepare your oven and pans. Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper.
Make the streusel topping first. In a medium bowl, combine the oats, flour, brown sugar, and cinnamon. Using your fingers or a pastry cutter, work the cold butter into the mixture until it resembles coarse crumbs. Place in the refrigerator while preparing the cookie dough.
Prepare the apples. Peel, core, and dice the apples into small 1/4-inch pieces. If your apple pieces are too large, they’ll release too much moisture during baking.
Mix the dry ingredients. In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg until well combined.
Cream the butter and sugars. In a large bowl, beat the softened butter, brown sugar, and granulated sugar with an electric mixer on medium speed until light and fluffy, about 2-3 minutes.
Add the wet ingredients. Beat in the eggs one at a time, then stir in the vanilla extract until fully incorporated.
Combine wet and dry ingredients. Gradually add the flour mixture to the butter mixture, mixing on low speed just until combined. Do not overmix or your cookies will be tough.
Fold in the diced apples gently with a spatula until evenly distributed throughout the dough.
Portion and top the cookies. Using a cookie scoop or tablespoon, drop rounded portions of dough (about 2 tablespoons each) onto the prepared baking sheets, spacing them about 2 inches apart. Sprinkle each cookie generously with the chilled streusel topping, pressing it lightly into the dough.
Bake until golden. Place in the preheated oven and bake for 12-14 minutes, or until the edges are lightly golden but the centers still look slightly underdone (they’ll continue cooking on the hot baking sheet).
Cool properly. Allow the cookies to cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.