In a medium saucepan over medium heat, melt the sugar, stirring constantly, until you get a dark amber caramel.
Add the butter and stir until completely dissolved in the caramel.
Pour in the milk and heavy cream, then stir continuously to dissolve the caramel in the liquid. Heat through, but do not boil.
In a separate bowl, whisk the egg yolks until light and fluffy. Temper the yolks by slowly drizzling in a small amount of the warm caramel mixture, whisking constantly. Then, pour the egg yolk mixture back into the saucepan.
Cook over low heat, stirring constantly, until the mixture thickens and coats the back of a spoon. This will take approximately 5-7 minutes.
Remove from heat, stir in the vanilla extract and a pinch of salt. Let it cool completely.
Pour the mixture into an ice cream maker and churn the ice cream according to the manufacturer's instructions.
When the ice cream is almost fully churned, add the crushed caramel pieces to give the ice cream a crispy texture.
Transfer the finished ice cream to a freezer-safe container and freeze for at least 3 hours to reach the desired consistency.
Before serving, let it sit at room temperature for a few minutes to make it easier to scoop.
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