Adjust an oven rack to the lower-middle position and preheat oven to 325 degrees F. Butter an 8-inch square baking pan. Line pan with extra long sheets of heavy duty foil so that there is an overhang after pressing foil into the corners/sides of pan. Overhang helps you to easily remove entire pan of brownies afterwards. Butter the foil bottom and sides. Set aside.
Whisk the flour, salt and baking powder together in a small bowl; set aside.
In a heavy saucepan, combine the chocolate and butter. Melt on low heat and constantly stir until completely melted and smooth. Whisk in the sugar and vanilla extract; allow the mixture to cool slightly. Whisk in the eggs, one at a time, fully incorporating each before adding the next. Continue to whisk the mixture is completely smooth. Add the dry ingredients, using a rubber spatula to fold until just incorporated.
Make the cream cheese filling by beating the cream cheese, sugar, vanilla and egg yolks together in a small bowl until evenly blended.
Pour half the brownie batter into the prepared pan, spreading it in an even layer. Gently spread the cream cheese mixture over the batter. Layering the remaining brownie batter over cream cheese filling.
Bake until the edges of the brownies have puffed slightly, the center feels not quite firm when touched lightly, and a toothpick or cake tester inserted 1-inch from the edge (not the center!) comes out with several moist, fudgey crumbs adhering to it, about 45 minutes. It will not seem fully done, but will set nicely upon cooling.
Cool the brownies in a pan on a wire rack until room temp, and then chill at least 3 hours. Remove brownies using foil overhang, transferring entire dessert onto a cutting board. Cut into squares and serve (a sharp serrated bread knife works well.)
Notes
The brownies can be stored in an airtight container in the refrigerate for several days and frozen for several weeks. Serve chilled for best results.