Combine the half-and-half and heavy cream in a medium saucepan over medium heat. Bring the mixture just to a simmer, stirring occasionally, and remove from the heat.
In a medium mixing bowl, whisk the egg yolks until they lighten in color. Gradually add the sugar and whisk to combine. Temper the cream mixture into the eggs and sugar by gradually whisking in small amounts, until about 1/3 of the cream mixture has been added.
Add the remaining cream to the egg mixture and return the entire mixture to the saucepan and place over low heat. Continue to cook, stirring frequently, until the mixture thickens enough to coat the back of a spoon and reaches 170 to 175ºF. Pour the mixture into a container and allow to sit at room temperature for 30 minutes. Add the peppermint oil and stir to combine. Place the mixture in the refrigerator. Once it is cool enough not to form condensation on the lid, cover and store for 4 to 8 hours or until it reaches 40ºF or below.
Pour the mixture into an ice cream maker. Add the chopped candies and process according to the manufacturers directions, 25 to 35 minutes. Serve as is for soft serve or freeze for another 3 to 4 hours to allow the ice cream to harden.