In a medium saucepan, bring the goat's milk and cream just to a boil; remove from the heat.
Meanwhile, in a medium stainless-steel bowl, whisk the egg yolks with the brown sugar. Very gradually whisk in 1/2 cup of the hot milk and cream, then gradually whisk in the rest. Return the mixture to the saucepan and cook over low heat, stirring constantly, until the custard thickens, about 10 minutes. Remove from the heat and stir in the salt and rum.
Strain the custard into a large stainless-steel bowl set in an ice-water bath and stir occasionally until thoroughly chilled. Working in 2 batches, freeze the custard in an ice cream maker according to the manufacturer's instructions. Freeze the ice cream in airtight containers for up to 1 day.