If using an ice cream machine, make sure you are following the machine instructions and freezing the cooling bowl for at least 12 hours before preparing your ice cream! The bowl has to be completely frozen for the ice cream to properly churn. Keep in mind if you are making both ice creams from scratch, it's best to make one ice cream the day before assembly of the cake to ensure that the ice cream maker stays frozen for both ice creams.
Start by making a custard. Mix cream, milk, sugar, and a pinch of salt in a small saucepan. Whisk eggs and vanilla together in a separate heatproof bowl.
Heat up the milk mixture until just below boiling.
Slowly stream half of the mixture into the eggs to temper the eggs, whisking constantly. Add the egg mixture into the saucepan, still whisking. Heat over medium to low heat until it boils for a few minutes, whisking the whole time to avoid scrambling the eggs. Once the mixture coats the back of a spoon and you can run a finger across the spoon and it leaves a trail, the custard is ready.
Strain your custard into a bowl and let cool. Once cool, cover with plastic wrap that touches the surface of the custard to avoid creating a skin, and refrigerate until cold.
When the custard is totally cool, start your ice cream machine and add the mixture to the cooling bowl (follow your machine instructions!). Ours took about 30 minutes to churn.
Once the ice cream is ready, line a cake tin (the same size as you baked the cake in) with plastic wrap and add all the ice cream to the tin, smoothing to create a disk. Place in the freezer until ready to assemble. (If using storebought ice cream, simply let your two ice cream flavors soften until it is easy to spread into the cake tin. Make two disks by lining the tin with plastic wrap, covering the ice cream, and freezing until ready to use.)