In a medium saucepan over medium heat, melt the butter. Add the sugar and stir continuously until it turns a dark golden brown.
Carefully pour in the milk and heavy cream, stirring constantly to prevent the caramel from crystallizing. Heat through, but do not boil.
In a separate bowl, whisk the egg yolks until light and fluffy. Slowly add a small amount of the warm caramel mixture to temper the eggs, whisking constantly.
Pour the entire mixture back into the saucepan and cook over low heat, stirring constantly, until it thickens. It's ready when it coats the back of a spoon.
Remove from the heat and stir in the vanilla extract and salt. Let it cool completely.
Once cooled, 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 shallow container and place it in the freezer. Every 30 minutes, stir with a fork to break up any ice crystals. Repeat this process for 3-4 hours.
Before serving, let it sit at room temperature for a few minutes to soften slightly.
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