Preheat oven to 350°F. Prepare the Batter: Coat a 13- x 9-inch baking dish with baking spray. Line bottom and sides of dish with parchment paper, leaving a 2- to 3-inch overhang on sides. Lightly coat parchment paper with baking spray.

Microwave butter and chocolate in a large microwavable bowl on HIGH until melted and smooth, 1 1/2 to 2 minutes, stopping to stir every 30 seconds. Whisk in sugar. Add eggs 1 at a time, whisking until just blended after each addition. Whisk in food coloring and vanilla.

Whisk together flour, cocoa, and salt in a separate bowl; whisk into butter mixture until just combined. Pour Batter into prepared baking dish.

Prepare the Cream Cheese Swirl: Beat egg whites, cream cheese, and sugar with an electric mixer fitted with a paddle attachment on medium speed until smooth, about 2 minutes. Add flour and vanilla; beat on low speed until combined, about 1 minute.

Drop Cream Cheese Swirl into Batter in heaping tablespoonfuls; swirl gently using a knife to create a marbled effect.

Bake in preheated oven until a wooden pick inserted in center of brownies comes out clean, 30to 35 minutes. Let cool completely in baking dish on a wire rack, about 1 hour.