Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13 inch baking pan.
Stir together the brownie mix, the 1/2 cup Irish cream liqueur, vegetable oil, and eggs. Spread in the prepared pan.
Bake in the preheated oven until the top is dry and the edges have started to pull away from the sides of the pan, about 30 minutes. Remove from oven to cool completely on a wire rack.
Place the milk, 2 tablespoons Irish cream liqueur, and coffee in a small, microwave safe bowl. Microwave on High until the mixture boils. Set aside to cool completely.
Beat the butter and confectioners' sugar with an electric mixer in a bowl until smooth. Add the vanilla extract and the cooled Irish cream mixture, and beat well. Spread evenly over the cooled brownies. Sprinkle evenly with the toffee bits.