Heat the milk, sugar, whole coffee beans, salt, and 1/2 cup of the cream in a medium saucepan, over medium heat, until it is warm and steamy, but not boiling. Once the mixture is warm, cover, remove from the heat, and let steep at room temperature for 1 hour.
Pour the remaining 1 1/2 cups of cream into a medium size metal bowl, set on ice over a larger bowl. Set a mesh strainer on top of the bowls. Set aside.
Reheat the cream and coffee mixture over medium-low heat again, until hot and steamy, but not boiling. In a large bowl, whisk the eggs. Temper the eggs by slowly ladling some of the cream and coffee mixture into the eggs while whisking the eggs vigorously to prevent them from scrambling.
Once you've incorporated about half of the cream and coffee mixture into the eggs, pour the custard back into the pan, stir to combine. Cook the custard over moderately low heat, stirring constantly, until a thermometer registers 170 degrees (Do not boil).
Pour the custard through the strainer and stir it into the cream. Press on the coffee beans in the strainer to extract as much of the coffee flavor as possible. Then discard the beans. Mix in the vanilla and finely ground coffee, and stir until cool.
Chill the mixture thoroughly in the refrigerator, then freeze it in your ice cream maker according to the manufacturer's instructions.