2 cups plus 2 tablespoons all-purpose flour
3/4 cup unsweetened cocoa powder
1 tsp baking soda
1/2 tsp salt
2 1/2 sticks butter, softened
2 cups granulated sugar
3 large eggs
2 1/2 tsp vanilla extract
8 ounces cream cheese, softened
1/2 cup confectioners' sugar
Preheat oven to 325 degrees. Coat a 9-by-13-inch baking dish with cooking spray. Line dish with parchment, leaving a 2-inch overhang on both long sides, then spray parchment.
Sift flour, cocoa powder, baking soda, and salt into a large bowl. In another bowl, cream butter and granulated sugar with a mixer until smooth. Add 2 eggs and 2 teaspoons vanilla, and beat until smooth, scraping down sides of bowl. Reduce speed to low; add flour mixture. Increase speed to medium; beat until incorporated.
Reserve 1 cup of dough; cover, and refrigerate. Press remaining dough into bottom of prepared dish. Refrigerate for 30 minutes. Bake until base is set and edges are puffed, 22-25 minutes (don't overbake or this brownie layer will be too hard and crumbly). Let cool in dish on a wire rack.
Mix cream cheese, confectioners' sugar, and remaining egg and 1/2 teaspoon vanilla in a medium bowl. Spread over cooled base; crumble reserved dough on top. Bake until filling is set, 25 to 30 minutes. Let cool in dish on wire rack, then lift out using overhang. Cut into 24 squares. Squares will keep, covered and refrigerated, for up to 3 days.