Grease a 10-inch fluted tube pan (such as a Bundt(R)) and coat inside of pan with unsweetened cocoa powder.
Beat cream cheese, confectioners' sugar, 1 egg, and Irish cream liqueur in a bowl until smooth and creamy. Set aside.
Beat chocolate cake mix, chocolate pudding mix, 3 eggs, and vegetable oil in a separate bowl until batter is thick and smooth. Slowly beat Irish stout beer into the batter.
Spoon 1/3 of the cake batter into the prepared pan; layer the Irish cream filling on top. Spoon remaining cake batter over the filling.
Bake in the preheated oven until a toothpick inserted into the cake comes out clean, 40 to 50 minutes.
Loosen the edges of the cake from the pan with a thin spatula; let cake rest in pan 15 minutes. Invert pan over a wire rack to release cake and allow to cool completely on the rack.
While cake is cooling, melt butter in a saucepan over medium heat and stir in white sugar and Irish whiskey. Bring to a boil, stirring to dissolve sugar, reduce heat to low, and simmer until thickened, about 10 minutes.
Transfer cake to a serving platter. Generously brush whiskey glaze over cooled cake several times and allow glaze to soak into cake.
