In a large mixing bowl, whisk the eggs. Add the buttermilk, sugar, cocoa powder, red food coloring, vinegar, vanilla, baking soda, and salt. Whisk well. Stir in the bread cubes and chocolate chips. Let sit for 30 minutes, stirring occasionally.
Preheat the oven to 325 degrees. Pour the mixture into the prepared baking dish and bake for 50-60 minutes or until set. Let sit for 10-15 minutes.
In the bowl of an electric mixer, combine the cream cheese, powdered sugar, and milk. Beat until smooth. Drizzle on top of the bread pudding while still warm.