• Preheat the oven to 350ºF (180ºC).
• Butter or spray an 8-inch square baking pan. 
• Line with a piece of parchment paper if you want, covering the bottom and two sides of the pan. The other two sides will remain unlined but greased. Or use two pieces, making a crisscross pattern. This will help you remove the brownies from the pan. 

For the brownie batter:

• Melt ½ cup unsalted butter in a small bowl and reserve. 
• Crack 1 egg into a small bowl, beat lightly, and pour half of it into a large bowl. The other half is used in the cheesecake batter below. 
• Add the remaining 2 eggs to the half in the bowl, and beat with ½ cup granulated sugar and 3 tablespoons brown sugar for 3 minutes, until light colored and foamy. They will triple in volume and make the brownies mousse-like.
• Add the melted butter and beat until smooth.
• Sift the dry ingredients (⅓ cup all-purpose flour, ⅛ teaspoon salt, and 5 tablespoons unsweetened cocoa powder) over the egg mixture.
• Add 1 teaspoon vanilla extract and mix with a spatula until smooth.
• Pour into the prepared pan. Reserve while you mix the cheesecake batter.

For the cheesecake batter:

• Beat together 6 ounces cream cheese, the remaining half egg from before, 2 tablespoons sugar, and 1 tablespoon all-purpose flour in a medium bowl until smooth and lump-free. Spoon the cheese mixture over the chocolate batter in mounds.
• Scatter ⅔ cup fresh or frozen raspberries on top and swirl with the tip of a knife. Don't overmix everything. You want the two batters to be seen. 
• Bake for about 20 minutes, until a tester or toothpick comes out with very moist crumbs attached. It should jiggle a bit in the center, like all brownies. That's how you ensure a fudgy brownie.
• Remove from the oven and cool completely on a wire rack. I like to keep them refrigerated for a day before eating, covered in plastic wrap. I think the flavor is much better. But you can eat them the same day, of course. 
• Run a smooth-bladed knife along the sides to unstuck the brownies. Lift carefully the paper to remove it more easily from the pan. Cut into squares and serve. 
• Refrigerate or freeze leftovers well-covered in plastic wrap or an airtight container.