In a saucepan over medium heat, combine the milk, sugar, and salt together. Heat the mixture, stirring until sugar is dissolved and mixture is smooth.

In a large bowl, whisk together the egg yolks. Add the hot milk mixture in a steady stream to the egg yolks, while whisking constantly.

Pour the mixture back into the saucepan and set over medium heat. Cook until the mixture is thickened and coats the back of a spoon, while stirring constantly, about 3 minutes.

Pour the mixture through a strainer into a large bowl. Stir in the heavy cream and peppermint extract. Chill completely in the fridge.

Freeze in an ice cream maker according to manufacturer's directions, adding the crushed cookies and candy cane pieces during the last 5 minutes of churning. Scoop ice cream into a large bowl or plastic container and freeze for 2 hours to firm up before serving.