Heat the oven to 350°F (175°C). Line a 9 x 13-inch metal baking pan with parchment paper, leaving enough overhang to lift out the brownies, and lightly grease the parchment.

Whisk the melted butter and granulated sugar together. Add the eggs one at a time, whisking after each addition, then whisk in the vanilla extract and Baileys Irish Cream.

In a separate bowl, sift or whisk together the all-purpose flour, unsweetened cocoa powder, and fine salt. Add the dry mixture to the wet mixture and fold just until no dry streaks remain.

Fold in the semisweet chocolate chips or chopped chocolate if using. Spread the batter evenly in the prepared pan.

Bake for approximately 25–32 minutes, until the edges are set and a toothpick inserted near the center comes out with moist crumbs. Do not wait for the toothpick to come out completely clean.

Cool the brownies completely in the pan. For especially neat, dense slices, refrigerate the cooled brownies for 30–60 minutes before frosting.

Beat the softened butter until smooth. Gradually beat in the powdered sugar, then add the Baileys, vanilla, and salt, beating until light and creamy; add heavy cream or milk only if the frosting is too thick.

Spread the frosting over the completely cooled brownies. Chill briefly if desired, then lift the brownies from the pan and cut them into squares.