• Chill the bowl of the ice cream maker in the freezer for 24 hours.

Prepare the hot chocolate mixture

• Chop up the bittersweet chocolate into fine pieces.
  
• Add the milk, heavy cream and sugar in a saucepan and heat until the sugar dissolves and the milk starts to simmer.
  
• Add the cocoa powder and chopped chocolate and whisk until smooth.
  
• Pour into a heat-proof measuring cup.

Prepare the egg mixture

• Place the egg yolks and sugar in the bowl of an electric mixer fitted with the paddle attachment. Beat on high speed for 3-5 minutes until light yellow and very thick.
• Bring the mixer down to low speed. Slowly pour the hot chocolate mixture into the egg mixture.

Cook the custard

• Pour the egg and chocolate mixture back into the saucepan. Cook over medium-low heat, stirring constantly, until thickened. Do not allow the mixture to boil, otherwise the egg yolk will get cooked. A candy thermometer will register about 180 degrees F.
• Pour the mixture through a sieve into a bowl and stir in the Kahlúa, vanilla, and salt.
  
• Place a piece of plastic wrap directly on top of the custard. Place in the refrigerator and chill completely, about 4 hours.
  

Churn and freeze the gelato

• Pour the custard into the bowl of an ice cream maker and process according to the manufacturer's directions (usually about 25 minutes).
  
• Pour into a covered container and chill in the freezer 8 hours to overnight.