Preheat oven to 350°F. Line two baking sheets with parchment.
Toss the diced apples with lemon juice, then blot away excess moisture with a paper towel.
Whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg in a medium bowl.
Cream softened butter, brown sugar, and granulated sugar until light and fluffy, about 2 minutes.
Beat in the egg and vanilla until smooth and combined.
Add the dry mixture to the bowl and mix on low just until no dry spots remain.
Fold in the apples until evenly dispersed; do not overmix.
Scoop rounded tablespoons of dough onto sheets, spacing about 2 inches apart. Gently flatten tops.
Bake 11–13 minutes, until edges are set and lightly golden.
Cool on the sheet 5 minutes, then transfer to a rack to cool completely.
Whisk powdered sugar, vanilla, milk, and a pinch of salt until smooth and pourable.
Drizzle glaze over cooled cookies and let set 15 minutes before serving.