Beat butter and sugars until fluffy. Beat in the egg, vanilla and apple butter.
In a separate bowl, whisk the flour, baking soda, cinnamon and salt.
Stir the flour mixture into the apple butter mixture until combined. Cover the dough and chill at least 30 minutes.
Preheat oven to 350 degrees. Using a 3 Tbsp. cookie scoop, place cookie dough 1 inch apart on a parchment-lined baking sheet. Bake 15 minutes. Cool completely before frosting.
Make the frosting: Beat the butter until fluffy. With the mixer on low speed, beat in the confectioner's sugar, vanilla bean paste and cinnamon. Store at room temperature.