The Raspberry Sauce: Combine raspberries and 2/3 cup sugar in a saucepan. Boil over medium heat, smashing berries as they cook. Simmer for 10 minutes until thickened. Strain through a fine-mesh sieve to remove seeds, then chill in the fridge.
The Chocolate Syrup: In a small pan, boil cocoa powder, coffee, and 1/2 cup sugar for 30 seconds. Off-heat, stir in the bittersweet chocolate until melted. Stir in vanilla and salt. Set aside.
Prepare the Slurry: Whisk the 2 tablespoons of milk with the cornstarch until smooth. Set aside.
Emulsify: Whisk the softened cream cheese into the warm chocolate syrup until no lumps remain.
Boil the Cream: In a large saucepan, combine 1 1/2 cups milk, heavy cream, the remaining 1/2 cup sugar, and corn syrup. Bring to a boil and cook for 5 minutes, stirring frequently.
Thicken: Remove from heat and whisk in the cornstarch slurry. Return to heat and cook for 1 minute until slightly thickened.
Chill: Combine the chocolate-cheese mixture with the hot cream mixture. Place the bowl in an ice water bath and stir every 5 minutes until cool. Refrigerate for at least 8 hours or overnight.
Churn: Process the chilled base in your ice cream maker according to the manufacturer's directions.
Layer and Swirl: In a freezer-safe container, layer 1/3 of the churned ice cream, 1/3 of the raspberry sauce, and 1/3 of the chocolate chunks. Repeat twice more.
Final Freeze: Freeze for at least 4 hours before serving to allow the ice cream to "ripen" and firm up.