In a large mixing bowl, whisk together the melted butter, brown sugar, and granulated sugar until smooth and combined. Add the cocoa powder and vanilla extract, stirring to incorporate fully. Crack in the egg and mix until the batter becomes glossy and thick.
In a separate bowl, sift together the all-purpose flour, baking soda, and salt. Gradually add the dry ingredients to the wet mixture, folding gently until just combined. Be careful not to overmix, as this will keep your cookies tender.
Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone mat. Scoop about 1 tablespoon of dough and flatten it slightly in your palm. Place a peppermint patty in the center, then carefully mold the dough around it, sealing the edges so the candy is fully enclosed. Place the cookie balls on the baking sheet, spacing them about 2 inches apart.
Bake the cookies for 10-12 minutes. They will look slightly underbaked on top, but that’s perfect for achieving a soft and chewy center. Remove from the oven and let them cool on the baking sheet for 5 minutes before transferring to a cooling rack.
While the cookies cool, combine the chocolate chips and heavy whipping cream in a small saucepan over low heat. Stir continuously until the chocolate is fully melted and smooth. Remove from heat and let it cool slightly.
Using a spoon or piping bag, drizzle the melted chocolate over the cooled cookies. Allow the drizzle to set for about 15 minutes before serving or storing.