In a saucepan over medium heat, caramelize the sugar until it turns golden brown. Be careful not to burn it.
Add the butter and stir until completely melted into the caramel. Carefully pour in the heavy cream and milk, stirring constantly to prevent the sugar from crystallizing.
In a separate bowl, whisk the egg yolks. Gradually whisk in a small amount of the warm caramel milk mixture to temper the eggs and prevent them from curdling.
Pour the egg mixture back into the saucepan and cook over low heat, stirring constantly, until the cream thickens and coats the back of a spoon.
Remove from heat, add the ground walnuts and vanilla extract, then mix well. Let the mixture cool completely to 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 it into a freezer-safe container and stir it every hour during the freezing process.
Leave it in the freezer for at least 4-5 hours to achieve the right consistency. Before serving, let it sit at room temperature for a few minutes.
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