Prep the pan and oven: Heat oven to 350°F (175°C).
Line an 8-inch square pan with parchment, leaving overhang for easy lifting. Lightly grease.
Melt chocolate and butter: In a heatproof bowl, melt butter and chopped chocolate together in the microwave in 20–30 second bursts, stirring between each, until smooth. Cool 5 minutes.
Whisk in sugars and flavor: Add granulated sugar, brown sugar, vanilla, and espresso powder to the warm chocolate mixture.
Whisk until glossy.
Add eggs: Whisk in the eggs one at a time until fully combined and slightly thickened.
Fold in dry ingredients: Sift in flour, cocoa powder, and salt. Gently fold with a spatula just until no dry streaks remain. Do not overmix.
Stir in chocolate chips if using.
Make the cheesecake mixture: In a separate bowl, beat cream cheese and sugar until smooth. Mix in egg yolk, vanilla, and brewed coffee (or milk) until creamy and pourable.
Layer and swirl: Spread about 2/3 of the brownie batter in the pan. Spoon the cheesecake mixture over the top in large dollops.
Add the remaining brownie batter in smaller spoonfuls between the cheesecake. Use a butter knife to gently swirl, dragging through the layers 3–4 times. Don’t overwork the pattern.
Bake: Bake for 28–35 minutes, until the edges are set and the center wobbles slightly but a toothpick inserted in a brownie area comes out with moist crumbs (not wet batter).
Cheesecake patches may puff a little.
Cool completely: Set the pan on a rack and cool to room temperature. For the cleanest slices, chill at least 2 hours or overnight.
Slice and serve: Lift out using the parchment. Warm a knife under hot water, wipe dry, and cut into 9–16 squares.
Enjoy slightly chilled or at room temperature.