Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper.
In a large bowl, beat the butter, ½ cup of granulated sugar, and brown sugar until creamy and light, about 2-3 minutes.
Add the egg and vanilla and beat again, scraping the sides of the bowl.
Add the flour, cocoa powder, baking soda, and salt, mixing until combined. It's a very soft dough, but I have found that it doesn't require chilling.
Use a small cookie scoop to scoop the dough into 24 balls. Roll them between the palms of your hands to make nice, round spheres, then roll them in the remaining sugar in a shallow dish to coat the outside of each cookie.
Place the balls of dough a couple inches apart on the prepared baking sheets and bake for 7-9 minutes, just until puffy and barely set, but very, very soft. Slightly underbaking is key to maintaining the soft texture of these cookies.
Wait 4-5 minutes before pressing a peppermint kiss into the center of each cookie, then wait another 2-3 minutes before using a spatula to transfer them to a wire cooling rack. Placing the peppermint kisses earlier will result in them melting into a puddle in the center of the cookies.