Make frosting: Place dark chocolate chips in a large bowl; set aside.

Combine 1 1/2 cups white sugar, water, and corn syrup in a saucepan. Heat until mixture is deep amber, about 15 minutes. Remove from heat and carefully pour in heavy cream. Place back on low heat, boil for 5 minutes, and pour over chocolate chips. Let stand for 1 minute.

Stir until chocolate is melted and smooth. Add vanilla, Irish cream, and butter 1 tablespoon at a time until incorporated. Cool to room temperature, about 4 to 6 hours.

Combine stout beer and 1 tablespoon dark brown sugar in a saucepan. Simmer until reduced to 1 cup, 8 to 10 minutes. Cool to room temperature, about 20 minutes.

Preheat oven to 350°F (175°C). Grease and flour 2 (9-inch) cakes pans.

Sift together flour, cocoa powder, baking powder, baking soda, and salt; set aside.

Beat butter, vegetable oil, brown sugar, and white sugar in a mixer until fluffy, about 4 minutes. Add eggs one by one, then add sour cream and vanilla extract.

Add half of dry ingredients and mix on low. Add reduced stout, then the remaining dry ingredients. Mix until just combined.

Divide batter into pans and tap to remove bubbles. Bake until cakes spring back lightly, about 30 to 35 minutes. Cool before removing from pans.