Preheat your oven to 350°F (175°C). Grease and flour a 9x5-inch loaf pan.
In a large mixing bowl, combine the flour, sugar, cocoa powder, baking powder, baking soda, and salt. Whisk until well blended.
Add the eggs, milk, vegetable oil, and vanilla extract to the dry ingredients. Mix on medium speed for 2 minutes until smooth.
Carefully stir in the boiling water until the batter is well combined. The batter will be thin.
Pour the batter into the prepared loaf pan and bake for 50-60 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.
While the cake cools, prepare the frosting. In a medium bowl, beat the softened cream cheese and butter together until creamy.
Gradually add the powdered sugar, mixing until smooth. Then, add the Bailey's Irish Cream and vanilla extract, mixing until well combined.
Once the cake is completely cool, spread the Bailey's cream cheese frosting evenly over the top.
Serve immediately or refrigerate for up to 3 days.