Brown the Butter: Melt butter in a light-colored pan over medium heat. Cook until it foams and develops brown specks and a nutty aroma. Remove from heat, stir in the semi-sweet chocolate until melted, and set aside.
Make Jam: Cook raspberries, sugar, and lemon juice in a small pan until berries break down. Stir in the cornstarch slurry and simmer until thickened. Let cool.
Cheesecake Mix: Whisk softened cream cheese, sugar, egg yolk, and vanilla until smooth. Transfer to a piping bag.
Egg Mixture: In a large bowl, whisk sugars, eggs, and vanilla for 5 minutes on high speed until the mixture is pale, thick, and reaches the "ribbon stage."
Combine: Fold the flour and cocoa powder into the egg mixture. Gently fold in the melted chocolate-butter mixture and white chocolate chunks.
Assemble: Pour batter into a lined 8-inch square pan. Pipe the cheesecake mixture and dollop the raspberry jam over the top. Use a skewer to create swirls.
Bake: Bake at 180°C (350°F) for 20–28 minutes. The center should still have a slight wobble.
Cool: Allow the brownies to cool completely in the pan before slicing into squares.