To make the cookies:

1. Beat butter and peanut butter for 30 seconds to incorporate. Add in granulated sugar followed by the light brown sugar beating until smooth and lump free, about 2 to 3 minutes.
2. Add in eggs one at a time, beating well after each addition. Add in the vanilla, baking powder, and baking soda.
3. Gradually add in the flour until a dough forms, scrape the sides and bottom of the bowl as needed.
4. Wrap the bowl with plastic wrap and refrigerate the dough for 1 hour.
5. Preheat heat to 375°F and line two baking pans with parchment paper, set aside.
6. Form the dough into small 3 to 4-inch cylindrical log-like shapes.
7. Flatten the logs and place them on the parchment-lined trays with 2-inches of space on each side.
8. Press the tines of a fork down on the cookie, pressing in one direction and then the other direction.
9. Use your fingers to squeeze the center to create the peanut shape.
10. Bake for 7 to 8 minutes or until the edges are lightly golden but not too dark. Let the cookies cool on the pans for 2 to 3 minutes before transferring them onto a wire rack to finish cooling completely.

To make the frosting:

1. In a clean bowl beat butter until smooth, light, and fluffy, about 2 to 3 minutes on a medium speed. Add in peanut butter, beating to combine it with the butter.
2. Gradually add in the powdered sugar followed by the vanilla. If the frosting looks too thick, add in a bit of milk or cream a tablespoon at a time.
3. Spread a tablespoon of frosting onto the backside of one cookie and then top it with another cookie, creating a sandwich. Repeat this process with the remaining cookies.