Preheat the oven to 350 degrees F. Lightly grease the bottom only of a 9-inch springform pan that has been lined with parchment paper. In a medium bowl, cream together the butter and sugar. Add the extract and eggs, beating well after each addition. Add the flour and cocoa and mix well to combine. The batter will be stiff. Spread in the pan. Set aside.
In a large bowl, cream the cream cheese and ricotta with the sugar until smooth. Add a small amount of the cheese mixture to the melted chocolate and combine well. Once the little bit of cheese mixture and chocolate is mixed well, add it back to the rest of the cheese mixture and beat until well combined. Add the eggs and flour and beat well. Smooth over the top of the cake batter in the pan. Bake until the cheesecake is set in the center, 25 to 30 minutes. Cover with foil during last 10 minutes to prevent over browning. Remove from the oven and cool.
A cheesecake is not like a regular cake. It is considered set when the center is still wobbly. After you chill the cheesecake, it will set without wobbling. At this point let the cake cool. Once it is cool, you need to chill the cake for an hour or two. Once your chocolate and cheesecake base has been chilled, you can start making the third layer. Beat the heavy whipping cream until stiff and set aside. In separate bowl, beat the cream cheese, sugar and vanilla with a mixer in large bowl until well blended. Fold the whipped cream into cheese mixture until just combined. Spread the mixture over the baked cheesecake layer and chill well (freezing the cake at this point works well and makes spreading the final layer easier). Make sure you chill for several hours to make sure all the layers are set.
Place the dark chocolate chips in a large bowl. Heat the cream and butter over low heat until just boiling. Reserve 2 tablespoons of the cream mixture. Pour the rest of the cream mixture over the chocolate and let set about 2 minutes, or until it begins to melt. Stir to combine. Repeat for the white chips using the reserved cream mixture. Let cool slightly until spreading consistency. Using a sharp knife, gently run the blade around the edge of the cake to loosen. Remove the cake sides from the pan and place the bottom with cake on a serving plate or stand. Spread the frosting over the top and sides. Drizzle the white chocolate lightly over the top. Using a knife or toothpick, run it lightly over the top to swirl to create desired effect. At this point, I would place the white chocolate ganache in a pastry bag and use a small round tip to decorate the top of the cake instead of trying to just drizzle it on top. You have more control. Press the mini chips into the frosting on sides.