In a large bowl, stir the melted butter with the brown sugar and granulated sugar until combined. Add the cocoa powder and stir until evenly incorporated.
Stir in the egg, vanilla, salt, and baking soda until combined.
Carefully stir in the flour until just mixed and no large streaks of flour remain. Do not overmix.
Line a baking sheet with parchment paper or a silicone baking mat. Use a 1-tablespoon scoop (or measure) to portion the dough into 1-tablespoon balls and place them on the prepared sheet. Spacing on the sheet does not need to be precise because the dough will be chilled.
Chill the cookie dough balls on the baking sheet in the refrigerator for at least 30 minutes. (If you chill longer than about 2 hours or freeze the dough, the cookies may spread less during baking.)
Preheat the oven to 350°F (177°C) while the dough finishes chilling.
Arrange the chilled dough balls about 2 inches apart on one or more lined baking sheets. If the dough was chilled longer than about 2 hours or frozen, gently press each dough ball with your palm to help it start spreading before baking.
Bake at 350°F for 8–12 minutes, or until the tops are no longer glossy and the edges look dry and set. The cookies should flatten as they bake.
Remove the cookies from the oven. Immediately press one unwrapped Peppermint Patty into the center of each hot cookie (use a spatula or a utensil if you prefer to avoid touching hot cookies). The peppermint patty will soften/melt against the warm cookie and then re-harden as it cools. Let the cookies cool on the baking sheet until the patties have partly re-set.
While the patties are setting, heat the 1/2 cup heavy whipping cream in the microwave or in a small saucepan until it is just about to boil (do not let it fully boil). Place the 1 cup chocolate chips in a medium heatproof bowl, pour the hot cream over the chips, let sit for about 30 seconds, then whisk until the mixture is smooth and fully combined (this is the ganache).
Spoon a little ganache over the top of each cookie, spreading it to cover the peppermint patty.
Let the ganache-topped cookies sit at room temperature for a few hours until the ganache is fully set. Once set, store the cookies in an airtight container and wait until they are fully set before stacking. Cookies may also be frozen.
Notes
There are no substitutions for heavy cream when making ganache.
You can also skip the ganache if you prefer.