Grease a 9"x13" pan and set aside. Preheat oven to 350 degrees F.
In a large mixing bowl, beat together butter, shortening and sugars until light and fluffy. Add egg, vanilla extract, and vinegar; mix well. Mix in baking soda, salt, and flour, just until blended. Stir in chocolate chips.
To make cheesecake filling, beat together cream cheese and sugar until well combined. Add egg and vanilla; beat until completely mixed.
In the prepared pan, spread half of the cookie dough. Pour cheesecake filling on top and spread evenly. With clean hands, take small amounts of the remaining cookie dough and flatten, placing them on top of the cheesecake filling.
Bake in preheated 350 degree F oven for 35 to 45 minutes, until golden brown and filling is set. Cool completely before storing in the refrigerator.