Preheat the oven to 375°F. In a large bowl, cream the butter, peanut butter and sugars until light and fluffy, about 4 minutes. Beat in the eggs and vanilla. In a separate bowl, whisk together the flour, baking soda, baking powder and salt. Gradually add the dry ingredients to the creamed mixture and mix well.
Shape the dough into 1-inch balls and place them 2 inches apart on ungreased baking sheets. Flatten each ball to about 3/8-inch thickness with a fork. Bake for 7 to 8 minutes or until golden, then remove to wire racks to cool completely.
To make the filling, combine the heavy cream, chocolate chips and vanilla in a large microwave-safe bowl. Heat on high for 1½ to 2 minutes, then stir until smooth. Let the mixture cool to room temperature; it will thicken as it cools. Spread the filling on half of the cookies and top with the remaining cookies to form sandwiches.