In a saucepan, combine the heavy cream, milk, and sugar. Heat over medium heat, stirring until the sugar is completely dissolved. Do not boil.
In a separate bowl, whisk the egg yolks. Gradually whisk in a small amount of the warm cream mixture to temper the eggs, preventing them from scrambling.
Pour the egg yolk mixture back into the saucepan and cook over low heat, stirring constantly, until the cream thickens enough to coat the back of a spoon.
Stir in the ground walnuts, vanilla extract, and rum. Mix well. The rum enhances the flavors and helps to keep the ice cream softer.
Let the mixture cool completely at room temperature.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions. If you don't have an ice cream maker, pour the mixture into a freezer-safe container and stir every hour during freezing to prevent ice crystals from forming.
Freeze for at least 4-5 hours, or until firm. Before serving, let it stand at room temperature for a few minutes to soften slightly.
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