Prepare the oven: Preheat the oven to 350°F and line a baking sheet with parchment paper.
Make the dough: Beat the peanut butter, softened butter, and brown sugar until light and fluffy. Mix in the egg and vanilla until smooth.
Add the dry ingredients: Add the flour, cornstarch, baking soda, and salt. Mix just until no dry streaks remain.
Shape the cookies: Scoop the dough into equal portions. Roll into balls, then flatten and pinch the center slightly to create a peanut shape. Arrange on the baking sheet and press crosshatches into both rounded ends with a fork.
Bake: Bake for 9–10 minutes, until the cookies are set around the edges but still soft in the center. Cool completely on a wire rack.
Make the filling: Beat the peanut butter, powdered sugar, heavy cream, and vanilla until smooth and creamy. Transfer to a piping bag.
Assemble: Pair up the cooled cookies. Pipe about 2 teaspoons of filling onto the flat side of one cookie and gently sandwich with the second cookie.