In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl using a hand mixer), beat butter, granulated sugar, and light brown sugar until light and fluffy.
Add eggs and vanilla extract, mixing until fully combined.
In a separate medium bowl, whisk together flour, cocoa powder, baking soda, baking powder, and kosher salt.
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Cover the dough with plastic wrap and refrigerate for at least 1 hour, or up to overnight.
When ready to bake, preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Using a 1-tablespoon cookie scoop, portion and roll dough into balls. Place on the prepared baking sheets about 2 inches apart.
Bake for 8-10 minutes, until the edges are set but the centers still look slightly soft.
Remove from the oven. Let the cookies cool completely on the baking sheets.
In a medium bowl, using a hand mixer, beat together peanut butter and butter until smooth.
Gradually add confectioners’ sugar and mix until light and fluffy.
Using a 1-tablespoon scoop, place a ball of the peanut butter mixture in the center of each cookie. Gently press down (I like to use a small piece of parchment paper so my hand doesn’t stick) so the filling spreads and covers most of the top, leaving just a small border around the edges.
Fill a pot about halfway full of water. Bring it to a simmer over low heat. Find a heat-safe bowl (glass or stainless steel) that can be set into the pot. Make sure the bottom of the bowl is not touching the water. The steam is what will melt the chocolate.
Add the chocolate and butter to the heat-safe bowl. With the heat on low, use a rubber spatula to stir the chocolate until it is melted and smooth. This takes about 5 minutes.
Spoon a small amount of the chocolate ganache over each peanut butter center, just enough to cover the filling.
Allow cookies to sit at room temperature until the chocolate sets, or chill briefly to help it firm up faster.
