Heat the milk, cream, and 1/2 cup sugar in a 2-quart saucepan, until the sugar dissolves and the milk starts to simmer. Add the cocoa powder and chocolate and whisk until smooth. Pour into a heat-proof measuring cup.
Place the egg yolks and the remaining 1/4 cup sugar in the bowl of an electric mixer fitted with the paddle attachment and beat on high speed for 3 to 5 minutes, until light yellow and very thick. With the mixer on low speed, slowly pour the hot chocolate mixture into the egg mixture. Pour the egg and chocolate mixture back into the 2-quart saucepan and cook over medium-low heat, stirring constantly, until thickened. A candy thermometer will register about 180°F Don't allow the mixture to boil!
Pour the mixture through a sieve into a bowl and stir in the coffee liqueur, vanilla, and salt. Place a piece of plastic wrap directly on top of the custard and chill completely.
Pour the custard into the bowl of an ice cream maker and process according to the manufacturer's directions. Stir in the roughly chopped chocolate, if using, and freeze in covered containers. Allow the gelato to thaw slightly before serving.