Preheat oven to 350 degrees F. Line 2 baking sheets with parchment paper, set aside.
In a medium bowl, whisk together flour, baking soda, baking powder and salt.
In a large bowl, beat butter, peanut butter and both sugars on high until light and fluffy. Stir egg and vanilla extract. Add dry ingredients to butter mixture, mixing until just combined. Stir in chocolate chips.
Using a small ice cream scoop, scoop cookie dough onto prepared baking sheets, allowing 2 inches of space between cookies. Bake until lightly golden brown around the edges, about 8-10 minutes. Remove from oven and let cool on baking sheets 5 minutes before transferring to wire racks to cool completely.
Once the cookies have completely cooled, turn half of them upside down. Add a scoop of ice cream to each and then sandwich together with the remaining cookies. Place on a baking sheet and freeze at least 1 hour or overnight. Wrap each individually in plastic wrap and keep in the freezer for up to 2 weeks.