Beat the butter with an electric mixer until pale and fluffy. Add the powdered sugar and beat for 2 minutes, until well combined. Add the vanilla and combine.
In a separate bowl, combine the flour and salt. Add the butter mixture and mix until the dough sticks together when pinched. Shape into a ball, wrap in plastic wrap and chill in the fridge until firm, about 1 hour.
Roll the dough out on a floured surface to 1/4 inch thick. Use a circular 2-inch round cookie cutter to cut out rounds. Press down in the center of the cookies to create a slight indentation. This will help the peanut butter stay put!
Place on a baking sheet lined with parchment paper. Bake at 325F for 13-15 minutes. Cool on the pan for 15 minutes, then transfer to a wire rack and cool completely.
Beat the peanut butter and powdered sugar with an electric mixer until smooth. Place into a piping bag fitted with a large, round piping tip.
Pipe some peanut butter on top of each cookie, trying to maintain a flat, smooth surface. Chill the cookies in the fridge for 30 minutes.
Dip the cookies in the melted chocolate using a fork. Allow the excess chocolate to drip off the cookies, then transfer to a baking sheet lined with parchment paper. Place in the fridge for 10 minutes, or until the chocolate has set. Enjoy!