Preheat the oven to 350 degrees F. Grease a 9-inch square baking pan with non-stick cooking spray and line with parchment paper.
For the brownies, place the butter and chocolate in a large heatproof bowl over a saucepan with 1 inch of simmering water. Use a spatula to stir occasionally until the butter and chocolate have melted. Remove the bowl from the heat.
Stir in 1-1/4 cup sugar, cocoa powder, 1 teaspoon vanilla extract and 1/2 teaspoon kosher salt.
Beat in 3 eggs until they are fully mixed into the brownie batter.
Then fold in the flour.
For the cheesecake swirl, use an electric mixer to beat the cream cheese, 1/2 cup granulated sugar, 1 teaspoon vanilla extract and 1 egg in a large bowl until smooth.
Alternate spoonfuls of brownie batter and cheesecake batter into the prepared baking pan. As you do this, sprinkle in half the Oreo cookies and reserve the rest for the topping.
Use a knife to swirl together the brownie and cheesecake batters. Do a figure-8 motion with some smaller swirls. Press the remaining Oreos on top.
Bake the brownies until they are just set in the center, but still very fudgy, about 30-35 minutes. When you insert a toothpick into the brownies, it should come out with some moist crumbs on it.
Let the brownies cool completely before removing them from the pan and cutting them into squares using a chef’s knife.