Bring milk just to a boil in a 1 1/2- to 2-quart heavy saucepan. Whisk together eggs, sugar, and a pinch of salt in a bowl, then add hot milk in a slow stream, whisking. Pour custard back into saucepan and cook over moderately low heat, stirring with a wooden spoon, until it registers 170 to 175°F on thermometer, 2 to 3 minutes. Immediately pour custard through a fine-mesh sieve into a clean bowl, then add cheeses and vanilla, stirring until cheeses are completely melted and incorporated. Cover surface of custard with a round of wax paper and chill until very cold, about 4 hours.

Freeze custard in ice cream maker. Transfer ice cream to an airtight container and put in freezer to harden, at least 2 hours.

Let soften at room temperature 20 minutes before serving.