Have all ingredients at room temperature. Butter and flour the wells of a mini cupcake pan. Set aside.
Preheat the oven to 350 F and position the rack in the bottom third of the oven.
Sift the flour, sugar, cocoa, baking soda, baking powder and salt into the bowl of an electric mixer. In a medium bowl, whisk together the melted butter, sour cream, eggs and vanilla until combined.
Pour the liquid ingredients into the mixer bowl with the dry ingredients and mix, using the flat beater attachment, on the lowest speed until the dry ingredients are moistened, 15 to 20 seconds. Turn the mixer off and scrape down the sides of the bowl. Increase the speed to medium and beat for 30 seconds. By hand, fold in the melted chocolate until combined.
Spoon the mixture into the mini cupcake pan, dividing it evenly among the 24 wells. Bake until a toothpick inserted in the center of a cupcake still has a few small crumbs attached to it, 10 to 12 minutes.
Let the cupcakes cool in the pan for 15 minutes before transferring them to a cooling rack. Let the cupcakes cool completely at room temperature, at least 1 hour, before serving.
Makes 24 mini cupcakes.