In a small bowl, sift the flour, baking soda, and salt together. Set aside.
Use a large bowl with a handheld mixer or a stand mixer to cream together the butter and brown sugar on medium speed, until the mixture is light and fluffy. Add the egg and vanilla extract and beat until well combined. Add the peanut butter and mix until just blended then reduce the speed to low and beat in the flour mixture just until the dough just comes together.
Cover the dough with plastic wrap and chill in the refrigerator until firm, about 1 hour.
When you're ready to bake them, preheat the oven to 350° F. Line 2 baking sheets with parchment paper. Fill a small bowl with the granulated sugar.
Roll the dough into logs that are 1 inch long by 1/2 inch wide. The easiest way to do this is to first roll the dough into 1 inch-wide balls, then keep rolling until they turn into cylinders. Roll each one in the bowl of sugar until covered.
Place the logs 2 inches apart on the baking sheets. Using the tines of a fork, create a criss-cross pattern in the dough and flatten the cookies until they are 1/4 inch high. Using your fingers, pinch the middle to create a peanut shape. Return the cookies to the freezer to cool for 15 minutes—the dough spreads quite a bit but cooling them before baking will help them retain their peanut shape.
Bake for 10 to 12 minutes, until lightly golden at the edges. Transfer to a wire rack. Meanwhile, make the filling.
For the peanut butter filling and assembly:
In a large bowl, use an electric mixer to beat the butter until smooth, about 2 minutes. Add the remaining ingredients and beat until well combined.
Pair each cookie with a similarly sized partner so that the edges match up. Spread a thick layer of filling on one cookie, then press the other cookie on top to create a sandwich. They are best eaten immediately, but will keep for up to 3 days in an airtight container.