Preheat the oven to 350 degrees F (175 degrees C). Lightly grease and line an 8x4-inch loaf pan with parchment paper.
Whisk together sugar and melted butter in a large bowl until smooth. Beat in eggs until fully combined. Stir in Irish cream liqueur and vanilla.
Fold in flour, baking powder, and salt just until combined; do not overmix. Stir in chocolate chips. Spread batter evenly into the prepared loaf pan.
Bake until a toothpick inserted into the center comes out clean, 40 to 45 minutes.
Cool loaf in the pan for 15 minutes, then remove from the pan and cool completely on a wire rack.
Prepare the frosting: Combine cream cheese, powdered sugar, Irish cream liqueur, and salt in a bowl. Beat with an electric mixer on low speed until smooth, adding milk as needed to make a spreading consistency. Spread frosting over the cooled loaf. Slice and serve.