Heat the oven to 350°F, lightly grease a 9-inch square baking pan, and line it with parchment paper. The parchment should cover the bottom and make lifting the chilled bars out of the pan much easier later.

Place the butter and 1 cup of chocolate chips in a large microwave-safe bowl and melt them in 20-second bursts, stirring after each round, until smooth and fully combined, about 2 minutes total. Stop as soon as the mixture is glossy and fluid so the chocolate does not scorch or separate.

Whisk the cocoa powder into the warm chocolate mixture, then add the granulated sugar, brown sugar, eggs, egg yolk, brewed coffee, and vanilla extract. The batter should look dark, thick, and shiny once everything is evenly mixed.

Fold in the flour and salt with a spatula just until no dry streaks remain, then scoop ½ cup of batter into a small bowl. Keep the mixing brief here so the brownie layer stays dense and fudgy instead of tough.

Stir the remaining ⅓ cup of chocolate chips into the main batter and spread it into the prepared pan. The layer will be thick, and smoothing it into the corners helps the base bake evenly.

Bake the brownie base for 20 minutes for a very fudgy result or 30 minutes for a more cakey result. While it bakes, stir the hot water into the reserved batter until it is still thick but slowly drips from a spoon like loose cake batter, adding a bit more water if needed.

While the base bakes, beat the cream cheese, sugar, and salt in a large bowl with a hand mixer or stand mixer on medium speed for about 2 minutes. The mixture should look very smooth, light, and free of stubborn lumps before the eggs go in.

Turn the mixer to low, add the eggs one at a time, scrape the bowl, and mix in the vanilla extract. Keep the speed gentle so you do not whip in excess air that can make the topping puff and crack.

Pour the cheesecake topping over the hot par-baked brownie base without cooling the pan first. The warm base helps the layers bond while still staying distinct.

Dollop the thinned reserved brownie batter over the cheesecake layer and swirl it with a skewer or the tip of a knife. Use a light hand so the pattern stays defined instead of blending into a muddy top.

Return the pan to the oven and bake for 10 minutes at 350°F. This short stretch gives the topping an early set before the lower finishing temperature.

Reduce the oven temperature to 325°F and bake for 20 to 25 minutes more, then cool the pan completely on a wire rack and refrigerate for 1 hour before slicing. The edges should be slightly puffed, and when you gently tap the pan, the center should wobble softly like set pudding rather than slosh.