Mix flour, baking soda, and salt.
In a separate large bowl, blend brown sugar, sugar, and butter with an electric mixer on medium speed, scraping down the sides of the bowl. Add eggs and vanilla, then beat at medium speed until light and fluffy.
Add the flour mixture and blend at low speed until just combined. Do not overmix.
Shape dough into marble-size balls. Place balls onto ungreased cookie sheet, about 1 inch apart. Bake 10 to 11 minutes at 300 degrees , but do not let them brown. Transfer cookies to a cool, flat surface.
Blend cream cheese and sugar on medium speed until fluffy. Add apple butter and beat until smooth and thoroughly combined. Spread 1 teaspoon of filling onto the flat side of a cookie. Top with another cookie to create a sandwich. Repeat with remaining cookies and filling. (Note: Fill cookies just before serving.)