Preheat oven to 350 degrees. Butter the bottom of a 10 1/2x15 1/2-inch jelly roll pan.
Place fudge brownie mix into a large bowl; beat in 1 cup of Irish cream liqueur, vegetable oil, and eggs until the mixture forms a smooth batter. Spread the batter into the prepared jelly roll pan.
Bake in the preheated oven until the brownies are set and a toothpick inserted into the center comes out clean, about 20 minutes. Remove pan from oven and allow to cool completely.
Beat unsalted butter in a large bowl until smooth; beat in 5 tablespoons Irish cream liqueur until mixture is creamy. Slowly beat in confectioners' sugar, 1 cup at a time, until frosting is desired stiffness; spread frosting on brownies and serve.