To Make the Cookies:

1. Preheat the oven to 350°F (175°C). Line a cookie sheet with parchment paper or a silicone baking mat. Pour 1/2 cup of sugar into a small bowl and set aside.
2. In a mixing bowl, whisk together the flour, baking soda, and salt. Set aside.
3. Using a standing mixer or electric hand mixer, cream together the butter, brown sugar, and peanut butter until smooth.
4. Add the egg and vanilla extract, and continue mixing until well combined.
5. Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
6. Divide the dough into small balls, about 1 inch in diameter. Shape each ball into a log, then roll in the reserved sugar.
7. Place the logs on the prepared cookie sheet, about 2 inches apart. Use the tines of a fork to make a criss-cross pattern on each log, then pinch the middle to create a peanut shape.
8. Bake for 10-12 minutes, or until the edges are lightly browned but the centers are still soft.
9. Allow the cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

To Make the Peanut Butter Filling:

1. Using a standing mixer, beat the butter on medium speed until fluffy.
2. Mix in the peanut butter, 1 cup of powdered sugar, milk, and vanilla extract until smooth.
3. Gradually add the remaining powdered sugar and mix until the filling is creamy and well combined.

Assembling the Cookies:

1. Pair up the baked cookies by size.
2. Spoon 1 Tablespoon of filling onto the bottom cookie of each pair.
3. Gently press the second cookie on top to create a sandwich.