four 9-inch cake boards; a 3-inch pastry ring; a 6-inch pastry ring; four 3-inch cake boards; four 6-inch cake boards
Preheat oven to 325 degrees F.
For the cheesecake layers: Spray two 9-inch round cake pans with cooking spray and line the bottoms with parchment circles. Spray the parchment. Place each cake pan into a larger cake pan to prepare for a water bath. Boil a kettle of water and have it at the ready.
Beat the cream cheese in a stand mixer fitted with the paddle attachment on medium speed for 2 to 3 minutes, stopping halfway through to scrape down the bowl and paddle. Add the granulated sugar and salt and blend on low, scraping the bowl when necessary, until the mixture is smooth and no lumps of cream cheese remain. Whisk together the eggs, yolks, sour cream and vanilla in a separate bowl. With the mixer on low speed, add the egg mixture to the cream cheese in 3 additions, allowing each addition to combine before adding the next. Give the bowl and paddle another scrape and blend just until smooth.
Divide the custard between the cake pans. Move the pans to your oven and carefully pour hot water into the larger cake pans, about 1 inch deep, to create a water bath. Bake the cheesecakes until the centers barely wobble and the internal temperature reads 160 degrees F, 1 hour 5 minutes to 1 hour 15 minutes. Turn off the oven and allow the cheesecakes to cool in the oven, 1 hour.
Remove the cheesecakes from the oven and cool for another hour at room temperature, then cover loosely and refrigerate until firm, 6 to 8 hours or up to overnight. Remove from the cake pans, transfer to cake boards and place in the freezer to firm up for 3 hours.
For the brownie layers: Preheat the oven to 325 degrees F. Spray two 9-inch round cake pans with cooking spray and line the bottoms with parchment circles. Spray the parchment.
Sift together the flour, baking powder and salt and set aside.
Combine the butter, unsweetened chocolate, vegetable oil, instant coffee and 1 cup of the mini chocolate chips in a double boiler over medium heat. Cook, stirring occasionally, until the chocolate has melted and the mixture is smooth. Remove from the heat and allow to cool.
Combine the brown sugar, eggs and vanilla in the bowl of a stand mixer fitted with the whip attachment, and whip at medium-high speed until lightened and thick, 3 to 4 minutes. Add the cooled chocolate and butter mixture and blend on low to combine. Add the flour and blend on low until it is nearly combined. Add the remaining chocolate chips and stir in using a spatula.
Divide the batter evenly between the cake pans. Bake until the brownies have puffed and a toothpick inserted into the centers comes out with sticky crumbs attached, 48 to 50 minutes. Allow to cool, then transfer to cake boards. Wrap in plastic and chill until cold and firm.
For the chocolate frosting: Melt the milk chocolate in a double boiler and allow to cool slightly.
Beat the cream cheese in a stand mixer fitted with the paddle attachment until soft and smooth, 2 to 3 minutes. Beat in the chocolate and scrape the bowl and paddle. Beat in the butter, salt and vanilla. Mix on high until light and smooth.
Use immediately. If the frosting stiffens it can be warmed by heating the mixer bowl with a kitchen torch, or by carefully holding the bowl over the stove for 10 seconds and then beating again until smooth.
For the ganache: Put the semisweet chips in a heatproof bowl and the white chocolate chips in another heatproof bowl. Heat the heavy cream in a saucepot over medium-low heat until just at a simmer. Pour 1 cup of the hot heavy cream over the semisweet chocolate chips, and pour the remaining 1/2 cup of hot heavy cream over the white chocolate chips. Let both mixtures sit until the chocolate has softened and begun to melt, 1 minute. Using rubber spatulas, stir each mixture until the chips have melted and the ganache becomes smooth. Let cool slightly before using.
To assemble: Cut the frozen cheesecake layers and the chilled brownie layers into three rings each using a paring knife and using 3-inch and 6-inch pastry rings or cake boards as guides (the smallest "rings" will truly be circles). Transfer the rings to cake boards and freeze the rings while you work.
Build the first layer on a cake stand by fitting together a large outer ring of brownie with a medium ring of cheesecake and a small circle of brownie. Top with a thin layer of chocolate frosting, then repeat these rings using a large cheesecake ring, medium brownie ring and small cheesecake circle. Top with a thin layer of frosting and repeat this pattern for the top two layers. Frost entire cake with a thin layer of chocolate frosting.
Pour a small amount of dark chocolate ganache in the center of the cake and let spread slightly. Pour a small amount of white chocolate ganache on top of the dark chocolate ganache, directly in the center of the cake. Let spread slightly, then pour another small amount of dark chocolate ganache in the center of the cake, creating rings. Repeat several more times, until the ganache has almost run to the top edge of the cake. Give the cake a gentle tap to let the ganache drip over the edge of the cake.
Chill until the ganache has set, about 20 minutes.