In a 2- to 3-quart pan, combine 1/2 cup butter and the unsweetened chocolate. Stir often over low heat until melted and smooth. Remove from heat and beat in granulated sugar, 3 tablespoons espresso powder, the eggs, and vanilla until well blended. Stir in flour just until incorporated.
Pour batter into a lightly buttered 8- or 9-inch cake pan with removable rim (see notes). Bake in a 350° regular or convection oven until center springs back when gently pressed, 20 to 25 minutes. Let cool completely in pan on a rack, about 1 hour.
In a bowl, combine remaining 1/4 cup butter and 1 tablespoon espresso powder with the powdered sugar and 2 tablespoons liqueur. With an electric mixer on medium speed, beat mixture until smooth and fluffy, adding up to 1 more tablespoon liqueur if necessary to make a smooth, spreadable mixture. Spread evenly over the top of the cool brownie cake.
Place chocolate chips in a 1-quart unpleated zip-lock plastic bag. Heat in a microwave oven at full power (100%) at 15-second intervals, squeezing chocolate between intervals, until melted and smooth, about 1 1/4 minutes total. Squeeze chocolate to one corner of the bag, then cut off about 1/4 inch of that corner. Pipe chocolate decoratively over buttercream. Chill until chocolate is firm, at least 30 minutes.
Just before serving, remove pan rim. Cut brownie cake into wedges and set on plates.