Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking pan or line it with parchment paper for easier removal.
In a medium saucepan over low heat, melt the butter and chopped semi-sweet chocolate together. Stir continuously until smooth. Remove from heat and let it cool slightly.
In a large mixing bowl, combine granulated sugar, brown sugar, and the melted chocolate mixture. Mix until fully incorporated.
Beat in the eggs one at a time, mixing well after each addition. Then, stir in the vanilla extract.
In a separate bowl, sift together flour, cocoa powder, baking powder, salt, and instant coffee granules. Gradually add this dry mixture to the wet mixture, stirring until just combined.
In another bowl, beat together softened cream cheese, powdered sugar, and milk until smooth and creamy.
Pour half of the brownie batter into the prepared baking pan, spreading it evenly. Then, dollop spoonfuls of the cream cheese mixture over the brownie batter and swirl gently with a knife. Pour the remaining brownie batter on top, spreading it to cover the cheesecake layer.
Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs (not wet batter).
Allow the brownies to cool in the pan for at least 15 minutes before cutting them into squares. Serve warm or at room temperature.
Notes
For best results, allow brownies to cool before cutting.