Whisk together eggs and salt in a large bowl, then add 1 cup hot cream mixture in a slow stream, whisking. Add vanilla. Pour into remaining hot cream mixture in saucepan and cook over moderately low heat, stirring constantly with a wooden spoon, until thick enough to coat back of spoon and registers 170 to 175°F on thermometer, about 5 minutes (do not let boil).
Pour custard through sieve into cleaned bowl and cool completely, stirring occasionally. Chill, covered, until cold, at least 3 hours.
Freeze custard in ice cream maker. Transfer ice cream to an airtight container and put in freezer to harden.