Set a fine-mesh strainer over a medium bowl. In a medium saucepan, whisk the granulated sugar, cornstarch, espresso, and salt. Whisk in the egg, then the milk and 1/4 cup Baileys. Bring to a boil over medium-high heat, whisking constantly, about 4 minutes. Boil 1 minute, whisking constantly. Remove from heat.

Whisk in the whiskey and butter. Pour into the prepared strainer and, using a heat-proof silicone spatula, press on the custard to strain. Scrape down the bottom of the strainer into the custard. Press a sheet of plastic wrap directly on top of the custard. Refrigerate until cold, about 3 hours.

Transfer the custard to a pastry bag or a resealable freezer bag. If using a freezer bag, snip off the tip of 1 corner. Pipe a large dollop of custard onto half of the doughnut holes; top with the remaining halves.

In a small bowl, microwave the chocolate in 15-second bursts until melted and smooth, about 1 minute. Whisk in the remaining 6 tablespoons Baileys for the chocolate sauce. Divide the doughnuts among 8 plates. Drizzle with the chocolate sauce. Dust with confectioners' sugar.