Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy.
Beat in the egg and vanilla extract until well combined.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Fold in the chocolate chips.
Drop tablespoon-sized balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes or until the edges are lightly golden.
Remove from the oven and let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
While the cookies cool, prepare the vanilla pudding by whisking together the pudding mix and milk in a bowl until thickened.
Once the cookies are cool, spread a layer of vanilla pudding on the bottom of half the cookies and top with the remaining cookies to create sandwiches.
Melt additional chocolate chips in the microwave or over a double boiler and drizzle over the top of the cookie sandwiches.
Allow the chocolate to set before serving.