Combine milk and half-and-half in a heavy saucepan over medium heat; cook to 180°. Remove from heat. Combine sugar, salt, and yolks, stirring well. Carefully pour 1 cup hot milk mixture gradually into yolk mixture, stirring constantly. Return mixture to pan. Cook custard over medium heat until a thermometer registers 160°, stirring constantly. Combine custard and 1 cup chestnuts in a medium bowl; cover and chill 8 hours. Strain mixture through a cheesecloth-lined strainer, pressing firmly; discard solids. Add vanilla.
Pour custard into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon ice cream into a freezer-safe container; cover and freeze 2 hours or until firm. Top ice cream with 1/2 cup chestnuts.