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Adjust the oven rack to the lower third position. Preheat to 350ºF (177ºC).
Lay down two foil sheets in an 8x8-inch baking pan perpendicular to one another, with extra foil hanging over the edges. Push the foil into the pan's corners and up the sides, and smooth out the surface. Lightly grease with baking spray or vegetable oil and set aside.
In a medium saucepan, fill about a third of it with water. Heat over low heat until barely simmering. Place a medium heatproof bowl over the top. The bottom of the bowl should not touch the water. Add the chocolate and butter to the bowl, occasionally stir until the mixture is smooth, about 5 minutes. Take the chocolate off the heat.
Brownie Layer: Whisk in the granulated sugar until smooth. Add the eggs, one at a time, whisking after each addition until combined, about 30 seconds per egg. Whisk in the vanilla extract until combined, about 10 seconds.In a medium bowl, whisk the flour, baking soda, and salt. Add the flour to the chocolate mixture in 3 additions. Gently fold with a spatula until the batter is just moistened, with flour streaks just disappearing. Do not overmix.
In a small bowl, add ½ cup of the brownie batter and 2 tablespoons of water, whisk to combine. This process makes it easier to swirl the topping later. Set aside. Transfer the remaining batter to the prepared pan, smoothing the surface.
Cheesecake Topping: In a stand mixer with the whisk attachment, add the cream cheese and mix on high speed (setting until smooth, scraping down the sides as needed, 1 minute.Add the granulated sugar and mix on medium-high speed (setting until combined, about 30 seconds. Whisk in the egg and vanilla extract on medium high speed (setting until combined, scraping down the sides of the bowl as needed, about 30 seconds.
Add the cheesecake topping over the brownie layer. Use a spatula to smooth the surface evenly.
Drop several 1 tablespoon-sized spoonfuls of the reserved brownie batter on top of the cheesecake. It takes about 9 drops to create a checkered pattern. Use a knife to make shallow swirls, creating a marbled pattern throughout. Evenly sprinkle chocolate chips on top.
Bake for 20 minutes, then loosely cover and tent the brownies with foil. It should not touch the surface. Rotate the pan and bake until a toothpick inserted in the center comes out with just a few moist crumbs attached, and the center reaches between 195 to 200ºF (91 to 93ºC), about 15 to 20 minutes.
Cool the brownies in the pan on a wire rack until they reach room temperature, about 1 ½ to 2 hours.
Use the foil to help lift and remove the brownies from the pan. Use a knife dipped in hot water and wiped down between each slice to give clean cuts. Cut into squares, 9 large or 16 smaller pieces. Store in an airtight container in the refrigerator for up to 3 days.