In a tall mixing bowl, use a hand blender to finely puree the blackberries and raspberries. Put a hair sieve over a bowl, rub the puree through there and set aside.
Beat the eggs with the sugar until thick and creamy.
Bring the milk and cream to a boil in a larger saucepan. Then turn the stove down and slowly, gradually, stir the mixture into the egg / sugar mixture. Then put this back into the pot and heat over a medium stove setting, while stirring, until it becomes thick. It is very important and careful not to let it boil. Then take it off the stove, put it in a freezer and let it cool down. Now stir in the vanilla extract and puree.
Now put in a freezer and let cool down. Close with the lid and put in the refrigerator for at least 4 hours, or overnight. Then process into an ice cream in an ice cream machine according to the manufacturer`s instructions and serve.