In a large mixing bowl, cream together the butter, shortening, sugars, vanilla, salt and vinegar. Beat in the egg, then the baking soda and flour. Stir in the chocolate chips. Divide the dough into two equal portions and set aside.
In a medium mixing bowl cream the cream cheese with the sugar until thoroughly combined. Beat in the egg and vanilla.
In a 9x13 pan, spread 1/2 of the cookie dough mixture evenly across the bottom. (This is somewhat tricky to do. I find it easier to add dollops of cookie dough throughout the pan and then spread all the dollops together. Spread the cream cheese mixture on the top of the cookie dough. With the remaining cookie dough, flatten chunks of dough between your hands into discs, and then place on top of the cream cheese mixture. (The dough discs don't have to touch, just place them near each other so they cover the surface as well as possible.).
Bake at 350 for 35-45 minutes, or until the top is brown and cooked through. Cool completely and store in the refrigerator.