Peppermint extract and crushed peppermint candies impart a refreshing taste to this ice cream. Delicious any time of year, the ice cream is an especially welcome treat when served at winter holiday gatherings.
In a medium saucepan over medium heat, warm 2 cups of the half-and-half and the corn syrup until steam begins to rise from the surface, 3 to 4 minutes. Remove from the heat.
In a heatproof mixing bowl, whisk together the egg yolks, sugar and salt until blended. Form a kitchen towel into a ring and place the bowl on top to prevent it from moving. Gradually add the hot half-and-half mixture, whisking constantly until fully incorporated. Return the mixture to the saucepan and place over medium-low heat. Cook, stirring slowly and continuously with a wooden spoon or spatula, until the custard thickens and a finger drawn across the back of the spoon leaves a path, 8 to 10 minutes; do not allow the custard to boil.
Pour the custard through a fine-mesh sieve set over a clean bowl. Stir in the peppermint extract. Add the remaining 1 cup half-and-half and stir to combine. Nestle the bowl in a larger one filled halfway with ice and water and cool the custard to room temperature, stirring occasionally. Refrigerate until chilled, at least 1 hour.
Transfer the custard to an ice cream maker and freeze according to the manufacturer's instructions. At the end of the freezing stage, add the peppermint candies and continue processing just until they are blended into the ice cream. Transfer the ice cream to a chilled container, cover and freeze until firm, 3 to 4 hours.