In a saucepan over medium heat, caramelize the sugar until it turns a golden brown color. Be careful not to burn it.
Add the butter and stir until completely melted into the caramel. Then, 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.
Pour the egg mixture back into the saucepan and cook over low heat, stirring constantly, until the custard thickens and coats the back of a spoon.
Remove from the heat, add the sea salt and vanilla extract, and stir well to combine. 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 the mixture into a freezer-safe container and stir every hour during the freezing process.
Leave in the freezer for at least 4-5 hours to achieve the desired consistency. Let stand at room temperature for a few minutes before serving.
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