Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper and set them aside.
In the bowl of a stand mixer, combine the softened butter, light brown sugar, and granulated sugar. Using the paddle attachment, cream them together on medium speed until the mixture becomes light and fluffy, about 3 minutes.
Add the creamy peanut butter to the mixture and mix for 1-2 minutes until fully incorporated.
Crack in the egg and add the vanilla extract. Mix for another 1-2 minutes until the mixture is smooth and clump-free.
In a separate bowl, whisk together the all-purpose flour, salt, and baking soda. Gradually add the dry mixture to the wet ingredients, mixing just until combined. Be sure not to overmix.
Using a tablespoon-sized cookie scoop, portion out the dough into small, even balls.
In a small bowl, place the remaining granulated sugar. Roll each dough ball in the sugar to coat it thoroughly before placing it on the prepared baking sheets. Ensure the dough balls are spaced about 2 inches apart.
Place the prepared cookie dough balls in the preheated oven and bake for 8 minutes, or until the cookies appear cracked and puffy.
After the first round of baking, remove the cookies from the oven. Gently press a Hershey's Kiss in the center of each cookie, being careful not to crack the cookie. The Kiss should sink slightly into the dough but remain on top.
Return the cookies to the oven and bake for an additional 2 minutes.
Remove the cookies from the oven and let them cool on the baking sheet for a few minutes. Then transfer them to a wire rack to cool completely.