Combine the butter and sugar in a medium-size saucepan over medium heat. When the butter has melted, stir to combine the melted butter with the sugar. Continue stirring while the mixture simmers for about 4 minutes, until thick, bubbling, and caramel colored. Remove from the heat.
Very slowly, pour the heavy cream into the hot caramel, while whisking constantly. Continue whisking until smooth, taking care to scrape the bottom of the pan as you stir the caramel. Add the milk, vanilla, and salt and whisk again until smooth. Refrigerate until cold.
When ready to churn the ice cream, pour the mixture into your ice cream maker and process it according to your machine's manufacturer's directions. When the mixture is almost done churning, add the crushed Heath bar pieces.
Serve immediately for a soft serve ice cream, or transfer to an airtight container and freeze for a few hours for firmer ice cream.