In a large saucepan, heat half-and-half cream to 175°; stir in sugar and salt until dissolved. Whisk a small amount of the hot mixture into the eggs. Return all to the pan, whisking constantly. Cook over low heat until mixture is just thick enough to coat a metal spoon and a thermometer reads at least 160°, stirring constantly. Do not allow to boil. Remove from heat immediately.
Quickly transfer to a large bowl; place bowl in a pan of ice water. Stir gently and occasionally for 2 minutes. Stir in cream and vanilla. Press plastic wrap onto surface of custard. Refrigerate several hours or overnight.
Fill cylinder of ice cream freezer; freeze according to the manufacturer’s directions. Transfer ice cream to freezer containers, allowing headspace for expansion. For each batch, stir in some of the peppermint candy. Freeze 2-4 hours or until firm.