In a large saucepan, heat the cream and milk to 175°; stir in sugar until dissolved. Whisk a small amount of hot mixture into the eggs. Return all to the pan, whisking constantly. Cook and stir over low heat until mixture reaches at least 160° and coats the back of a metal spoon.
Remove from the heat. Whisk in cream cheese until smooth. Cool quickly by placing pan in a bowl of ice water; stir for 2 minutes. Stir in lemon juice and vanilla. Press plastic wrap onto surface of custard. Refrigerate for several hours or overnight.
Fill cylinder of ice cream freezer two-thirds full; freeze according to manufacturer’s directions. Refrigerate remaining mixture until ready to freeze. Transfer to a freezer container; freeze for 2-4 hours before serving.