Preheat Oven and Prepare Pan: Preheat the oven to 350°F (175°C). Line an 8 × 8 × 2-inch baking pan with parchment paper, leaving a 1-inch overhang on two opposite sides to help lift out the brownies later. Lightly coat the parchment paper with nonstick cooking spray and set the pan aside. Cream Butter and Sugars: In a large mixing bowl, combine the room-temperature unsalted butter, granulated sugar, and lightly packed light brown sugar. Beat with a hand mixer or a stand mixer fitted with the paddle attachment for about 3 minutes, until the mixture looks light and fluffy. Add Eggs and Vanilla: Add the 2 large eggs and 1 teaspoon vanilla extract to the butter-sugar mixture. Mix until the ingredients are fully combined and the batter looks smooth.
Mix the Brownie Batter: Add the all-purpose flour, salt, and unsweetened cocoa powder. Mix until the dry ingredients are incorporated and the brownie batter is smooth and evenly blended.
Partially Bake the Brownie Base: Transfer the brownie batter to the prepared baking pan and spread it into an even layer. Bake for 12–15 minutes, until the brownie layer is mostly set. Remove the pan from the oven while preparing the cheesecake topping.
Prepare the Cheesecake Mixture: In a medium mixing bowl, combine the room-temperature cream cheese and granulated sugar. Beat with a hand mixer until smooth and creamy. Scrape down the sides and bottom of the bowl as needed to eliminate any lumps and ensure the mixture is evenly blended. Incorporate Egg and Vanilla: Add the large egg and ½ teaspoon vanilla extract to the cream cheese mixture. Beat again until the cheesecake filling is smooth and fully combined.
Spread the Cheesecake Layer: Spoon the cheesecake mixture over the partially baked brownie base. Use a rubber spatula to gently spread it into an even layer. Work carefully to avoid disturbing or pulling up the delicate brownie layer underneath.
Add Strawberry Swirls: Drizzle 2 tablespoons of strawberry sauce or strawberry jam over the cheesecake layer. Using the tip of a butter knife, gently swirl the strawberry topping through the cheesecake to create a marbled pattern. Avoid pressing the knife down into the brownie base. Bake Until Set: Return the pan to the 350°F (175°C) oven and bake for another 20 minutes, allowing the cheesecake layer to bake and set.
Cool the Brownies: Remove the pan from the oven and let the brownies cool on the counter for about 20 minutes before refrigerating.
Chill and Slice: Place the cooled pan in the refrigerator and chill for 2–3 hours, until the brownies are firm enough to cut. Lift the brownies from the pan using the parchment overhang, then cut into 9 brownies and serve.
