Preheat oven to 350°F. Line two baking sheets with parchment paper.
Stir diced apple with lemon juice and cornstarch until evenly coated; set aside.
Make the crumble: Combine oats, crumble flour, crumble brown sugar, and crumble cinnamon. Cut in cold butter until clumpy. Chill.
Cream softened butter with granulated sugar and brown sugar until light and fluffy, about 2 minutes.
Beat in applesauce and vanilla until smooth.
Whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Add to wet ingredients and mix just combined.
Fold in the coated apple pieces until evenly dispersed.
Scoop 1½-tablespoon mounds onto sheets, spacing 2 inches apart. Gently flatten tops.
Top each mound with a spoonful of chilled crumble and press lightly so it adheres.
Bake 12–14 minutes, until edges are set and lightly golden. Cool 5 minutes, then move to a rack.
Whisk powdered sugar, cider or milk, and cinnamon until smooth. Drizzle over cooled cookies.