In a medium saucepan, combine the milk, heavy cream, and honey. Heat gently over medium heat, stirring occasionally, until the mixture is steaming but not boiling.
Chop the dark chocolate and add it to the warm milk mixture. Stir continuously until the chocolate is completely melted and the mixture is smooth and homogenous.
In a separate bowl, whisk the egg yolks until they are pale and creamy.
Slowly pour a ladleful of the warm chocolate milk mixture into the egg yolk mixture, whisking constantly to temper the eggs. This prevents the eggs from scrambling.
Pour the tempered egg mixture back into the saucepan with the remaining milk mixture. Cook over low heat, stirring constantly, until the custard thickens and coats the back of a spoon (about 5-7 minutes). Do not let it boil!
Remove from the heat and stir in the vanilla extract. Let the mixture cool completely to room temperature. Then, transfer to the refrigerator and chill for at least 4 hours, or preferably overnight, to allow the flavors to meld.
If using an ice cream maker, pour the chilled mixture into the machine and churn according to the manufacturer's instructions. If not, pour the mixture into a freezer-safe container and freeze, stirring with a fork every 30 minutes for the first 2-3 hours to prevent ice crystals from forming and ensure a creamy texture.
Before serving, let the ice cream sit at room temperature for a few minutes to soften slightly, making it easier to scoop. Garnish with chocolate shavings or a drizzle of extra honey, if desired.
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