Grease 2 9-inch round cake pans and heat oven to 315 degrees F.
Sift together the flour, baking soda, baking powder, and salt. Add the milk, vinegar, sugar, oil and vanilla and stir until combined--do not overmix.
Divide batter equally into two bowls. Stir cocoa powder into one bowl and half the chocolate chips into the other bowl. Pour each bowl of batter into a prepared cake pan. Sprinkle the rest of the chocolate chips over the vanilla cake.
Bake for 40-50 minutes, or until the top of the cake springs back slightly when pressed. Remove and let cool completely. Remove layers from the pans and freeze, well-wrapped in plastic wrap, until ready to use.
For the ice cream:
Prepare an ice bath. Combine the milk and sugar in a medium saucepan. Remove about 1/2 cup of the milk and whisk in the cornstarch until smooth. Whisk cornstarch mixture back into the pot.
Bring mixture to a low boil, whisking constantly. Boil for 1-2 minutes, then remove heat and place in an ice bath. Whisk in the vanilla and rum. Let mixture chill fully (about 1.5 hours) or chill overnight before freezing in an ice cream maker.
To make the cake crumbs:
In a frying pan over medium heat, crumble about 3/4 cup of the chocolate chip cake into the pan and toast for 5-10 minutes until the crumbs start to turn golden. Let cool completely before use.
To make the ganache:
Place chocolate chips and milk in a microwave-safe dish. Microwave for 45 seconds. Stir until melted.
Stamp out a 6-inch circle of chocolate cake from the 9-inch layer, reserving the cake scraps. Line a 6-inch cake pan with a large sheet of plastic wrap.
Place all the scraps from the chocolate layer on the bottom of the cake pan, pressing down gently to form a flat layer. Add a scoop or two of softened ice cream, then the rest of the chocolate chip cake, breaking it up to form a second layer of cake. Add another scoop or two of ice cream on top, flattening with a spatula. Top with the 6-inch circle of chocolate cake. Spread ganache across the top and top with the toasted cake crumbs. Freeze for at least four hours before serving.
*You can use nearly any alcohol--this is just what I had on hand. The original recipe called for vodka, which is virtually flavorless..