1. Preheat the oven to 350°F. Grease an 8-inch baking pan with butter or nonstick cooking spray.

For the Cheesecake Batter

1. In a small bowl, microwave the cream cheese until soft, 20 to 30 seconds. In the bowl of an electric mixer, beat the softened cream cheese with the sugar, egg yolk, vanilla, Baileys and flour until smooth. Set aside.

For the Brownie Batter

1. In a medium bowl, melt the butter in the microwave, about 1 minute. Immediately add the chopped chocolate and stir until melted and smooth. (If bits of chocolate remain, place the mixture back in the microwave for 20 seconds.)
2. In the bowl of an electric mixer, beat the eggs, granulated sugar, brown sugar, vanilla, Baileys and salt on medium speed until smooth. Add the chocolate/butter mixture and beat until well incorporated. Add the flour and cocoa powder and mix on low until just combined. Do not overmix.
3. Transfer ½ cup of the brownie batter to a small microwave-safe bowl (it's fine to use the same bowl you melted the butter and chocolate in). Spread the remaining brownie batter in the prepared pan. Spread the cheesecake batter evenly over the brownie batter. Microwave the reserved ½ cup of brownie batter until warm and pourable, 15-20 seconds. Using a spoon, dollop the warm batter over the cheesecake layer, about 8 dollops (for the prettiest marble effect, leave as much space as possible between the dollops). Using a knife, swirl the brownie batter through cheesecake batter, making a marbled pattern. Don't overdo the marbling or the batters will get muddled.
4. Bake until puffed, set in the center and just starting to brown around the edges, 40-45 minutes. (Don't undercook, or the brownies will be too wet in the center.) Let cool completely in the pan on a wire rack. Cut the brownies into 12 or 16 squares, wiping the knife clean between slices. The brownies will seem very moist and fudgy at first, but will firm up once cut and left out for a bit. (The brownies may also be served warm with vanilla ice cream; just let them cool at least 30 minutes after baking.)
5. Freezer-Friendly Instructions: The brownies can be frozen for up to 3 months. After they are completely cooled, double-wrap them securely with aluminum foil or plastic freezer wrap. Thaw overnight on the countertop before serving.