Put the goat's milk, heavy cream, 1/4 cup of the sugar, and the chopped chocolate into a heavy-bottomed pot. Bring to a simmer, whisking the chocolate so that it melts and incorporates into the milk. Turn off the heat.
In a medium-sized bowl, whisk together the egg yolks and 1/2 cup of sugar. Whisk a cup of the hot milk mixture into the egg yolks and sugar. This will temper the eggs and help prevent them from curdling (turning into scrambled eggs) in the hot milk. Whisk in another cup and then pour that mixture back into the pot with the remainder of the milk mixture.
Turn the heat back on medium-low and cook, stirring, until the mixture begins to thicken. Remove from heat and strain through a fine-mesh strainer into a bowl. Whisk in the vanilla.
Cool thoroughly in the refrigerator, stirring occasionally until cold. This can be done the day before you want to make ice cream to ensure the mixture is ice cold before going into the ice cream machine.
Freeze according to your ice cream machine's directions.
*I used pasteurized goat's milk which is why I think the flavor was so mild.