Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
In the bowl of your electric mixer, beat the butter and peanut butter until creamy and combined. Beat in the sugars on medium speed until fluffy. Add in the vanilla extract. Beat in the flour on low speed until just combined. Scoop out the dough (it will be crumbly) and form it into two tubes with your hands. They should be about 2 inches in diamter and about 6 to 7 inches long. Wrap them in plastic wrap and refrigerate for 30 to 60 minutes.
Remove the dough from the fridge and cut 1/4 to 1/2 inch slices off the tube. Place on the parchment. Sprinkle each with salt. Bake for 20 minutes, or until just slightly golden. Let the cookies cool completely before dipping them in chocolate.
To melt the chocolate for dipping, microwave it (or heat over double boiler) for 30 seconds at a time on 50% power. Stir after each time until it just melts. Once melted, stir in the coconut oil. Dip an edge of each piece of shortbread and if desired, sprinkle with more salt.