FOR THE CREAM CHEESE FILLING: Microwave cream cheese until soft, 20 to 30 seconds.
Add sour cream, sugar, and flour and whisk to combine. Set aside.
Adjust oven rack to middle position and heat oven to 325 degrees.
Make foil sling for 8-inch square baking pan by folding 2 long sheets of aluminum foil so each is 8 inches wide. Lay sheets of foil in pan perpendicular to each other, with extra foil hanging over edges of pan. Push foil into corners and up sides of pan, smoothing foil flush to pan. Grease foil.
FOR THE BROWNIE BATTER: Whisk flour, baking powder, and salt together in bowl and set aside.
Microwave chocolate and butter in bowl at 50 percent power, stirring occasionally, until melted, 1 to 2 minutes.
Whisk sugar, eggs, and vanilla together in medium bowl.
Add melted chocolate mixture (do not clean bowl) and whisk until incorporated.
Add flour mixture and fold to combine.
Transfer 1/2 cup batter to bowl used to melt chocolate.
Spread remaining batter in prepared pan.
Spread cream cheese filling evenly over batter.
Microwave bowl of reserved batter until warm and pourable, 10 to 20 seconds.
Using spoon, dollop softened batter over cream cheese filling, 6 to 8 dollops.
Using knife, swirl batter through cream cheese filling, making marbled pattern, 10 to 12 strokes, leaving 1/2-inch border around edges.
Bake until toothpick inserted in center comes out with few moist crumbs attached, 35 to 40 minutes, rotating pan halfway through baking.
Let cool in pan on wire rack for 1 hour.
Using foil overhang, lift brownies out of pan. Return brownies to wire rack and let cool completely, about 1 hour. Cut into 2-inch squares and serve.