First, make the filling: In a medium bowl, combine the softened butter, light brown sugar, peanut butter, pure vanilla extract, confectioners' sugar, and salt and use a handheld electric mixer to cream the mixture together. The mixture will be thick and doughy, like a cookie dough. Set aside. Meanwhile, line 2 muffin tins with about 16 paper liners.
Melt the chocolate candy coating according to package directions, or until melted and smooth. Spoon about 1 Tablespoon of the melted chocolate into the bottom of each paper liner and use the back of the spoon to kind of brush it slightly up the sides of the paper liner.
Take a heaping Tablespoonful of the peanut butter filling and form it into a little disc. Place the disc into the bottom of the chocolate cup. Top the peanut butter disc with another Tablespoon of the melted chocolate and smooth it over the top of the disc in an even layer.
Allow the peanut butter cups to set and firm up, about 30 minutes at room temperature, or expedite the process by placing the peanut butter cups in the fridge. Store any leftovers at room temperature or in the fridge or freezer.