Make the cupcakes: Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with paper liners. Combine the cocoa powder and 1/4 cup water in a small bowl until smooth; set aside. Put the unsweetened chocolate in a microwave-safe bowl and microwave, stirring every 30 seconds, until melted; set aside and let cool.
Sift the flour and baking soda into a large bowl; stir in the salt. Beat the butter and granulated sugar in a separate large bowl with a mixer on medium speed until smooth. Beat in the eggs, one at a time, until combined. Reduce the mixer speed to low and add the cocoa mixture and the melted chocolate; mix until combined. Beat in the flour mixture in two batches, alternating with the milk in two batches, until combined. Beat in the liqueur.
Divide the batter among the prepared muffin cups, filling them three-quarters of the way. Bake until a toothpick inserted into the centers comes out clean, 20 to 25 minutes. Let cool 10 minutes in the pan, then remove the cupcakes to a rack to cool completely.
Meanwhile, make the ganache: Combine the semisweet chocolate, 1 cup heavy cream and the salt in a heatproof bowl set over a pan of simmering water; stir until melted and smooth, about 8 minutes. Remove the bowl from the saucepan and set aside until the ganache is cool and thick (but not hard), about 1 hour.
Beat the remaining 1/3 cup heavy cream and the confectioners' sugar in a large bowl with a mixer until soft peaks form. Fold in 1/4 cup of the cooled ganache until smooth. Cut a small hole (about 1/2 inch deep) into the center of each cupcake with a paring knife; remove the cake. Use a small measuring spoon to fill the hole with the chocolate whipped cream.
Beat the remaining ganache with a mixer on medium speed until light and fluffy, 3 to 5 minutes. (Be careful not to overbeat or the ganache will become grainy.) Transfer the whipped ganache to a pastry bag fitted with a star tip and pipe onto the cupcakes. Top with shaved white chocolate.