Preheat oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking pan.

Heat water in a small saucepan until simmering. Place chocolate in a bowl set over, but not touching, the water. Stir chocolate frequently until melted, 2 to 3 minutes. Remove from heat and discard water.

Sift flour, baking soda, and salt together in a bowl.

Beat 2 cups sugar and shortening in a large bowl with an electric mixer until creamy. Beat in 4 egg yolks until smooth. Stir in milk and vanilla extract. Mix in flour mixture. Fold melted chocolate into the batter.

Beat egg whites in another bowl until stiff peaks form, about 5 minutes. Fold egg whites into the batter. Pour half of the batter into the prepared baking pan.

Beat cream cheese and 1 1/2 cup sugar in a large bowl until smooth. Add 4 eggs; beat on low speed just until combined. Pour mixture over batter in the baking dish. Gently spoon remaining batter on top and spread it to the edges of the dish.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 75 to 80 minutes. Cool in the pan for 30 minutes, then transfer to a wire rack to cool, about 30 minutes more.