Place the half-and-half and the heavy cream into 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 adding small amounts, until about 1/3 of the cream mixture has been added. Pour in the remainder and return the entire mixture to the saucepan and place over low heat. Continue to cook, stirring frequently, until the mixture thickens slightly and coats the back of a spoon and reaches 170 to 175 degrees F. 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 into the refrigerator and once it is cool enough not to form condensation on the lid, cover and store for 4 to 8 hours or until it reaches 40 degrees F or below.
Pour the mixture into an ice cream maker; add the chopped candies and process according to the manufacturers directions. This should take approximately 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.