In a saucepan, whisk together 1 cup of the cream with the milk, egg yolks and sugar. Add 1/2 cup of the crushed candy canes, along with the salt and vanilla.
Over medium heat, cook the custard while whisking constantly, just until the custard is hot, beginning to gently simmer around the edges, and the candy canes are melted. Do not let the custard boil.
Remove from the heat and pour through a fine mesh strainer to remove any bits of cooked eggs.
Cover the custard and refrigerate until well chilled, about 6 hours or overnight.
When you're ready to churn your ice cream, use an electric mixer to whip the remaining 1 cup of heavy whipping cream until soft peaks form. Whisk the whipped cream into the chilled peppermint custard.
Churn the custard in your ice cream maker according to the manufacturer's instructions, until it has a soft-serve consistency. During the last minute of churning, add the peppermint schnapps or vodka.
Fold the remaining 1/2 cup crushed candy canes into the churned ice cream.
Transfer the ice cream to lidded containers and freeze until firm enough to scoop, at least 8-10 hours, or overnight.