Obviously this is a recipe focusing on North American paw paws, but I am pretty sure banana would work just as well, although the flavors would be very different. Fresh or frozen paw paw mash will work, so you can make this anytime if you have some squirreled away. Use really ripe paw paws here -- the blackish ones. This is essentially a French vanilla ice cream with paw paw added right at the end. The only tricky part of this recipe is tempering the egg yolks, and that's not exactly rocket science. You just need to add the hot cream in small doses while stirring the beaten yolks. I'd eat this ice cream within a few weeks. The longer it sits in the freezer, the hard it gets, and it can get icy over time.
Start by making the custard for the ice cream. Heat the cream and milk and sugar in a pot over medium heat to the steaming point, about 165°F. If you are using a real vanilla bean, add it to the mixture now. If you are using extract, wait a bit.
Beat the egg yolks in a bowl. Stirring the eggs all the time, add one ladle of the hot cream mixture into the eggs. Do this a second time -- this is tempering the eggs so they don't curdle in the hot cream. Pour the egg mixture into the pot.
Stir the custard often and heat it back to the steaming point. When it thickens -- it should coat the back of a spoon -- turn off the heat and pour the custard into a bowl. I like to set the bowl into a larger bowl filled with ice to cool the mix down quickly. Stir in the vanilla extract if that's what you are using.
When the custard is cool, whisk in the mashed paw paws until they are well combined. You can put the mixture into your ice cream maker now, or you can push it through a fine-meshed strainer to remove any stray bits; you'll need to fish out the vanilla bean if you used that anyway. Run through your ice cream machine and eat!
Notes
Note that prep time does not include chilling time after you make the custard.