In a medium saucepan, combine the half & half, heavy cream and half of the sugar. Bring to a simmer, stirring until the sugar is dissolved. Remove the pan from heat and stir in the chopped chocolate and cocoa powder; stirring until chocolate is melted and the mixture is smooth.
In a large bowl, use an electric mixer to beat the egg yolks with the remaining sugar until thick and pale. Add a small amount of the hot chocolate mixture, beating constantly. Add a little more. Keep adding slowly until you have warmed up the egg mixture a little bit-- eventually adding all of the chocolate mixture to the eggs. Transfer the gelato base back to the saucepan, stir in the espresso and cook the base over low to medium heat, stirring constantly, until a thermometer registers 170 degrees F (do not let the mixture boil). Pour the base into a bowl, cover and refrigerate several hours (or overnight) until very cold.
Pour the gelato base into an ice cream maker- following the manufacturer's instructions. It usually takes 25 to 30 minutes of processing before the base turns into a beautiful, creamy, frozen gelato. Transfer to an airtight container and store in the freezer to freeze completely.
Notes
If you are preparing this recipe as gluten free, just be sure to use a brand of chocolate that is known to be GF.