CUSTARD BASE: Beat eggs, sugar, honey and salt in medium heavy saucepan until blended; stir in milk. Cook over low heat, stirring constantly, until mixture is just thick enough to coat a metal spoon with a thin film and temperature reaches 160°F, about 15 minutes. Do not allow to boil. Remove from heat immediately.
Cool quickly. Set pan in larger pan of ice water; stir occasionally and gently for a few minutes to hasten cooling. Press piece of plastic wrap onto surface of custard to prevent a “skin” from forming. Refrigerate until thoroughly chilled, at least 1 hour.
ICE CREAM: Pour chilled custard, whipping cream and vanilla into 1-gallon ice cream freezer can. Freeze according to manufacturer’s directions, using 6 parts crushed ice to 1 part rock salt. Transfer to freezer containers, allowing head space for expansion; freeze until firm.
