Prep the Pan and Oven: Heat oven to 350°F (175°C). Line an 8-inch square pan with parchment, leaving overhang to lift out later. Lightly grease the parchment.

Bloom the Espresso: Stir instant espresso powder into hot water until dissolved. This step deepens the coffee flavor and mixes more evenly into the batter.

Melt Chocolate and Butter: In a heatproof bowl over barely simmering water (or in short microwave bursts), melt butter and chopped chocolate. Stir until smooth and glossy. Let cool for 3–4 minutes so it doesn’t scramble the eggs.

Whisk in Sugars and Flavor: Add granulated sugar, brown sugar, vanilla, and the dissolved espresso to the melted chocolate. Whisk until combined and slightly thick.

Add Eggs: Whisk in the eggs, one at a time, mixing just until shiny and cohesive. Avoid overbeating; you don’t want too much air in the batter.

Fold in Dry Ingredients: Sift in flour, cocoa powder, and salt. Fold gently with a spatula until no dry streaks remain. The batter will be thick and glossy.

Make the Cheesecake Mixture: In a separate bowl, beat cream cheese and sugar until smooth and lump-free. Add egg yolk, vanilla, sour cream, and espresso powder (if using). Mix until silky and spreadable.

Assemble: Spread about 2/3 of the brownie batter in the pan. Dollop spoonfuls of the cheesecake mixture over it, then add the remaining brownie batter in small scoops between the cheesecake dollops.

Swirl: Using a butter knife or skewer, gently swirl the batters together. Aim for broad S-shaped curves. Don’t overmix or you’ll lose the marbled look.

Optional Toppings: Sprinkle mini chocolate chips over the top and add a light pinch of flaky sea salt for contrast.

Bake: Bake 28–35 minutes. The edges should be set, and the center should still have a slight jiggle. A toothpick inserted 2 inches from the edge should come out with a few moist crumbs.

Cool Completely: Let the pan cool on a rack until room temperature, then chill for at least 1 hour before slicing. This step helps the cheesecake layer firm up and gives cleaner cuts.

Slice and Serve: Lift the slab out using the parchment. Warm a knife under hot water, wipe dry, and cut into squares. Wipe the blade between cuts for neat edges.