Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking pan or line it with parchment paper for easy removal.

In a medium saucepan, melt the butter over low heat. Once melted, remove from heat and stir in the granulated sugar and brown sugar until well combined.

Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.

In a separate bowl, whisk together the flour, cocoa powder, salt, and baking powder. Gradually add the dry ingredients to the butter mixture, stirring until just combined.

Pour half of the brownie batter into the prepared baking pan, spreading it evenly.

In a mixing bowl, beat the cream cheese until smooth. Add the granulated sugar, egg, and vanilla extract, mixing until well combined.

Spread the cheesecake mixture over the brownie layer in the pan. Drop spoonfuls of the remaining brownie batter on top of the cheesecake layer and use a knife or toothpick to swirl the two layers together gently.

Bake for 30-35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs. Allow to cool completely in the pan before slicing into squares.