Brownie Base

1. Set the oven temperature to 350°F. Prepare a 9-inch square baking pan by greasing it lightly with butter or baking spray and lining it with parchment paper.
2. In a large bowl suitable for the microwave, combine the butter and 1 cup of chocolate chips. Heat in 20-second intervals, stirring in between, until completely melted and mixed, which should take about 2 minutes. Incorporate the cocoa powder, then add the sugars, eggs, egg yolk, coffee, and vanilla, whisking until everything is well blended.
3. Mix in the flour and salt using a spatula until just combined. Set aside ½ cup of the brownie batter in a separate small bowl. Fold the remaining ⅓ cup of chocolate chips into the rest of the batter, then spread it evenly in the prepared pan.
4. Bake for 20 minutes for a fudgy texture or 30 minutes for a cakier result. While baking, stir hot water into the reserved batter until it remains thick but can pour slowly from a spoon. Add more water if necessary.

Cheesecake Topping

1. While the brownie base is baking, prepare the cheesecake topping by mixing the cream cheese, sugar, and salt in a large bowl. Use a hand mixer or a stand mixer with a whisk attachment on medium speed to beat until the mixture is very smooth and light, which should take about 2 minutes.
2. Lower the mixer speed and add the eggs one at a time. Scrape down the sides of the bowl, then mix in the vanilla. Carefully pour the cheesecake mixture over the brownie base that has been partially baked (no need to let it cool).
3. Dollop the reserved brownie batter over the cheesecake layer. With a skewer or knife tip, gently swirl the brownie batter into the cheesecake mixture.
4. Continue baking for an additional 10 minutes.
5. Lower the oven temperature to 325°F and bake for another 20 to 25 minutes, or until the cheesecake topping is puffed at the edges and the center has a slight wobble. Allow the pan to cool completely on a wire rack. Chill in the refrigerator for 1 hour before slicing.