Preheat the oven to 350°F (175°C). Line a 9-inch round (or 8-inch square) cake pan with parchment paper. Grease the paper and any exposed parts of the inside of the pan.
Whisk together the flour, baking powder, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter and sugar until light and fluffy. Add the eggs, one at a time, mixing well after each addition. Mix in the Irish cream and vanilla.
Reduce mixer speed to low. Add the flour mixture in three portions, alternating with two portions of milk. Mix just until combined.
Transfer the batter to the prepared pan and spread evenly. Tap the pan gently a few times on the counter to get rid of any large air pockets.
Bake the cake 35 to 40 minutes, or until the cake is golden brown and a pick inserted into the center comes out clean.
Place the pan on a wire rack and allow the cake to cool.
To make the frosting:
Using an electric mixer on medium speed, beat the butter until fluffy.
Gradually add the confectioners' sugar, and mix well.
Add the Irish cream, vanilla, and salt. Mix at medium-high speed until the frosting is smooth and fluffy. If needed, add a little bit of milk to the frosting to get it to a smooth, spreadable consistency.
Spread the frosting on the cooled cake. Add any garnishes you like.
Notes
Frosting recipe adapted from Bake from Scratch.
Store tightly covered in the refrigerator up to 5 days. Bring to room temperature for serving.