In a small saucepan, cook the milk, sugar and salt over medium-high heat, stirring occasionally, until the sugar dissolves, about 4 minutes (do not let the milk boil).
Slowly pour one-half of the hot milk mixture into the eggs, whisking. Pour the mixture back into the saucepan and cook over low heat, stirring and scraping with a heatproof spatula, until thick enough to coat the spatula, 8 to 10 minutes.
Strain the mixture into a metal bowl. Stir in the heavy cream. Refrigerate until cool, about 2 hours.
Stir the coffee liqueur and vodka into the ice cream base. Pour into an ice cream maker and process according to the manufacturer's instructions.