Place the saucepan over medium-low heat and warm, whisking frequently for 10-12 minutes, until the custard coats the back of a spatula. Scrape along the bottom periodically to make sure the bottom of the pan doesn’t scorch.
Pour the custard through a mesh strainer into a large bowl. Cover well and chill for 4 hours, or overnight.
Stir in the peppermint and churn the ice cream mixture in an ice cream machine according to the manufacturer’s instructions. Once the ice cream reaches the soft-serve consistency, add the crushed candy pieces and churn until incorporated. If desired, drip in 4-5 drops of natural red food coloring and churn until it’s just swirled in.