To make the cheesecake mixture, combine cream cheese and sugar in a large bowl, using a wooden spoon to cream together until smooth. Stir in the flour.
In a separate medium bowl, whisk together the egg, sour cream, vanilla extract, and lemon juice. Add this mixture gradually into the cream cheese mixture. Whisk until smooth. Set aside.
Prepare a shelf in the center of the oven and preheat to 350 degrees F. Prepare a 15″ x 15″ square of heavy duty aluminum foil. Turn over your 9″ square baking pan and center the aluminum foil over it so that it and down the sides so that it follows the shape of the pan. Remove the foil (maintaining shape) and turn over the pan, placing the aluminum foil on the inside. Smooth bottom and sides to the pan.
To make the brownie batter, strain together flour, baking powder, and salt in a small bowl. Set aside.
Melt the butter in a medium sized bowl placed over a pot of simmering water. The bottom of the bowl should not come in contact with the water. (This is called a double boiler.) When the butter is melted, add the dark and unsweetened chocolate and stir for 1 minute. Remove from heat and stir occasionally until chocolate is completely melted. Set aside.
Whisk the eggs in a large bowl. Continue whisking as you add the sugar in a steady stream. Whisk in the chocolate mixture. Stir the vanilla in last.
Strain the flour mixture over the chocolate mixture in three additions, folding gently each time. Remove 1 cup of the brownie batter and set aside.
Pour the remaining brownie batter into the prepared pan and level with a spatula. Pour the cream cheese mixture over the top and smooth into an even layer. Drop the reserved cup of brownie batter on top in about 8 spoonfuls. Swirl the layers together by inserting a small sharp knife halfway in and drawing an S pattern, as illustrated here. (Don’t make fun of my illustration skills please.)
How to swirl
Bake the brownies for 45-to-50 minutes, until it looks puffed and lightly browning. Cool on a cooling rack for 3-to-4 hours before cutting. Cut into 16 pieces. Serve.