Combine the half-and-half, vanilla bean, vanilla seeds and salt in a nonreactive saucepan over medium heat. Bring just to a boil, then immediately remove the saucepan from the heat.
In a large bowl add the sugar and egg yolks and whisk to combine. Add the half-and-half mixture, about 1/4 cup at a time, to the beaten eggs and sugar, whisking after each addition. Pour the mixture back into the saucepan and add the mint leaves. Cook over medium-low heat until mixture is thick enough to coat the back of a spoon, 5 to 6 minutes. Remove from the heat and strain through a fine mesh sieve into a clean bowl. Transfer the bowl to an ice bath and stir frequently until custard is completely cool. Refrigerate until thoroughly chilled.
Pour the custard into an ice cream machine and follow the manufacturer's instructions for churning time. As the ice cream begins to set, sprinkle in the crushed peppermint candies. Transfer the ice cream to a freezer-safe container and freeze until firm, about 8 hours.