Preheat oven to 325 degrees F (160 degrees C) and place the rack in the center of the oven. Have ready a 9 x 9 inch (23 x 23 cm) square baking pan that has been lined with aluminum foil across the bottom and up two opposite sides of the pan. Set aside.
In a stainless steel (heatproof) bowl placed over a saucepan of simmering water, melt the butter and chocolate. Remove from heat and stir in the sugar and vanilla extract. Add the eggs, one at a time, beating well (with a wooden spoon) after each addition.
Stir in the flour and salt and beat, with a wooden spoon, until the batter is smooth and glossy and comes away from the sides of the pan (about one minute). Remove 1/2 cup of the brownie batter and set aside. Place the remainder of the brownie batter evenly into the prepared pan.
Then in the bowl of your food processor (or with a hand mixer), process the cream cheese until smooth. Add the sugar, vanilla, and egg and process until creamy and smooth. Spread the cream cheese filling over the brownie layer. Spoon small dollops of the reserved brownie batter evenly on top of the cream cheese filling. Then with a table knife or wooden skewer, swirl the two batters without mixing them.
Bake in the preheated oven for about 25 minutes or until the brownies start to pull away from the sides of the pan and the edges of the brownies are just beginning to brown. Remove from oven and place on a wire rack to cool. Refrigerate the brownies until they are firm enough to cut into squares (at least two hours).
Once chilled, remove the brownies from the pan by lifting with the ends of the foil and transfer to a cutting board. With a sharp knife, cut into 16 squares. It is a good idea to have a damp cloth nearby to wipe your knife between cuts. These brownies can be stored in the refrigerator for several days.
Makes about 16 – 2 inch squares.