Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone baking mat. Set aside the granulated sugar in a small bowl for rolling the dough.

In a medium mixing bowl, whisk together the flour, baking soda, and salt. Set aside.

In a large mixing bowl, using a stand mixer or handheld electric mixer, cream the softened butter, brown sugar, and peanut butter until smooth and fluffy.

Add the egg and vanilla extract to the creamed mixture and mix until fully combined.

Gradually add the dry ingredients to the wet mixture, mixing on low speed until just incorporated. Avoid overmixing to maintain a tender texture.

Roll the dough into 1-inch balls, then form each ball into a small log shape. Roll the logs in the reserved granulated sugar and place them on the prepared baking sheet, spaced about 2 inches apart.

Use a fork to press a crisscross pattern on each dough log, then gently pinch the middle to form a peanut shape.

Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden but the centers remain soft. Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.

In a clean mixing bowl, beat the softened butter until fluffy. Add the peanut butter, powdered sugar, milk, and vanilla extract, mixing until smooth and creamy.

Pair the cooled cookies into similar-sized groups. Spread 1 tablespoon of the peanut butter filling onto the bottom side of one cookie and top with another cookie to form a sandwich. Repeat with the remaining cookies and filling.