To make the crumb topping
1-Combine the flour, brown sugar, cinnamon, and: Combine the flour, brown sugar, cinnamon, and salt in a medium bowl.
2-Add the cold butter, and mix with: Add the cold butter, and mix with a pastry blender, a fork, or your fingers until the mixture forms crumbs.
3-Refrigerate while you make the cookies: Refrigerate while you make the cookies.

To make the cookies
4-Preheat the oven to 375°F (190°C): Preheat the oven to 375°F (190°C). Line baking sheets with parchment paper or silicone liners.
5-Whisk together the flour, baking soda, cinnamon,: Whisk together the flour, baking soda, cinnamon, nutmeg, and salt. Set aside.
6-Using an electric mixer on medium speed,: Using an electric mixer on medium speed, beat the butter, brown sugar, and sugar until light and fluffy. Add the egg and vanilla, and mix well. Mix in the apple butter.
7-Reduce the mixer speed to low: Reduce the mixer speed to low. Gradually add the flour mixture, mixing just until combined or a few small streaks of flour remain.
8-Stir the apples into the dough: Stir the apples into the dough.
9-Drop the dough by 2-tablespoon portions onto: Drop the dough by 2-tablespoon portions onto the prepared pans, leaving about 2 inches between the cookies.
10-Top each cookie generously with the crumb: Top each cookie generously with the crumb topping, pressing down gently to adhere.
11-Bake one pan at a time for: Bake one pan at a time for 10 to 12 minutes, or until the cookies and topping are golden brown.
12-Cool the cookies on the pans on: Cool the cookies on the pans on wire racks for 5 minutes. Then transfer the cookies from the pans directly to wire racks to cool completely.