Preheat oven to 350°F (180°C). Spray a 13x9-inch baking pan with cooking spray. Line pan with parchment paper, letting excess extend over sides of pan.

For swirl: In the bowl of a stand mixer fitted with the paddle attachment, beat cream cheese at medium speed until smooth. Add granulated sugar and flour; beat until well combined, stopping to scrape bottom and sides of bowl. Add eggs, one at a time, beating just until combined after each addition and stopping to scrape sides of bowl; beat in extracts. Cover and refrigerate.

For batter: In the top of a double boiler, combine chocolate and butter. Cook over simmering water, whisking frequently, until melted and smooth. Remove from heat; whisk in sugars until smooth.

In a medium bowl, whisk together flour, cocoa, and salt.

Add eggs, one at a time, to chocolate mixture, whisking until combined after each addition; whisk in vanilla. Fold in flour mixture until just combined. Spread two-thirds of batter (about 4⅓ cups or 1,097 grams) into prepared pan.

Using an offset spatula, gently spread swirl mixture onto batter in pan. Dollop remaining batter onto swirl mixture. Using a wooden skewer or knife, gently swirl layers together.

Bake until edges are set and a wooden pick inserted in center comes out with a few moist crumbs, 55 minutes to 1 hour, covering with foil during final 30 minutes of baking to prevent excess browning. Let cool completely in pan on a wire rack. Cover and refrigerate for at least 2 hours or up to overnight.

Using excess parchment as handles, remove from pan, and cut into bars. Refrigerate in an airtight container for up to 3 days.