Preheat your oven to 350°F (175°C). Line your baking sheets with parchment paper to prevent sticking.
In a large mixing bowl, cream together the softened butter, sugar, and powdered sugar until light and fluffy. Add in the vegetable oil and mix until fully incorporated.
Beat in the eggs one at a time, mixing well after each addition.
In a separate bowl, whisk together the cornstarch, cream of tartar, baking soda, salt, cocoa powder, and flour. Gradually add this dry mixture to the wet ingredients, mixing until just combined.
Using a cookie scoop, portion out the dough and roll into balls. Roll each ball in the additional 1/4 cup of sugar before placing them on the prepared baking sheets. Bake for 10-12 minutes, or until the edges are set. Allow to cool slightly on the baking sheet before transferring to a wire rack to cool completely.
In a small bowl, mix together the creamy peanut butter, melted butter, powdered sugar, and vanilla until smooth.
Once the cookies are cool, take one cookie, spread a generous amount of the peanut butter filling on the flat side, and then top with another cookie to create a sandwich. Repeat until all cookies are filled.
In a medium saucepan over low heat, combine the chopped chocolate or chocolate chips with the heavy cream. Stir until the chocolate is fully melted and the mixture is smooth.
Dip each cookie sandwich into the melted chocolate, ensuring they are fully coated. Place the coated cookies back onto the parchment-lined baking sheets and refrigerate until the chocolate is set.