With a knife, halve vanilla beans lengthwise. Scrape seeds into a large heavy saucepan and stir in heavy cream, milk, sugar, and salt. Bring mixture just to a boil, stirring occasionally, and remove pan from heat. Add peppermint extract.
In a large bowl lightly beat egg yolks. Add hot cream mixture to eggs in a slow stream, whisking, then return custard mixture to pan. Cook custard over moderate heat, and do not let it come to a boil. Stir constantly until mixture reaches 170 degrees (F).
Pour custard through a sieve into a clean bowl and cool. Chill custard, its surface covered with wax paper, in refrigerator until cold, about three hours.
Add cold mixture to ice cream maker and proceed according to manufacturer's instructions. Just before ice cream is set, add crushed candy to ice cream and continue mixing until combined.
Place ice cream in an airtight container and freeze until fully set, at least two hours. Enjoy!
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