Beat eggs, sugar, and salt in large heavy saucepan until blended. Stir in 2 cups milk. Cook over low heat, stirring constantly but gently, until mixture is just thick enough to just coat a metal spoon with a thin film and temperature reaches 160 degrees, about 15 minutes.
Do not allow to boil. Remove from heat immediately. Stir in remaining 2 cups milk and vanilla. Refrigerate, covered, until thoroughly chilled, several hours or overnight.