Making the Ice Cream:

1. In a 2 to 3-quart 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.

2. 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, about 8 to 10 minutes. Do not allow the custard to boil.

3. Pour the custard through a fine-mesh sieve set over a clean bowl. Add 1/4 cup of the peppermint candies and stir until melted. 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.

4. 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 remaining 1/4 cup candies and continue processing just until they are blended into the ice cream.

5. Transfer the ice cream to a chilled container, cover and freeze until firm, 3 to 4 hours.