In the bowl of a stand mixer or in a large bowl with a hand mixer, cream together butter and sugar until light and fluffy, 3-4 minutes.
Add egg, vanilla, and salt and beat until smooth.
Turn mixer to low and add flour. Mix until just incorporated, taking care not to over-mix.
Place dough on a large piece of plastic wrap. Fold plastic wrap over dough to completely cover and use your hands to roll the dough into a smooth log, about 2" in diameter. Place in fridge to chill until firm, about 2 hours.
Preheat oven to 350 degrees and line a baking sheet with parchment paper.
Remove dough from fridge and use a sharp knife to slice log into ¼"-thick rounds. Arrange cookies on prepared baking sheet about 1" apart. You should get about 60 rounds.
Bake for 9-12 minutes, until edges are just starting to turn golden. Let cookies cool in pan for a couple minutes before moving to wire racks to cool completely.
While cookies are baking and cooling, make the peanut butter filling. In the bowl of a stand mixer or in a large bowl with a hand mixer or rubber spatula, combine peanut butter and confectioners' sugar and mix until well combined.
Spread a layer of peanut butter filling over each cooled shortbread cookie.
Put chocolate in a double boiler (or in a heat-proof bowl set over a saucepan filled with water) and bring water in the bottom of boiler to a boil. Stir the chocolate with a wooden spoon until it's evenly melted and smooth.
Put peanut butter topped cookies in melted chocolate. Use two forks to flip cookies to coat entirely with chocolate. Remove and set on piece of waxed paper or parchment paper and let chocolate harden.