For the ganache glaze: Bring the cream to a boil in a small saucepan. Combine the chocolate and corn syrup in a small heatproof bowl. Pour the hot cream over the chocolate and let stand until melted, about 5 minutes. Whisk until smooth. Let the glaze cool until thick but still pourable, about 5 minutes.
For the doughnuts: Pour the oil into a large saucepan set over medium heat. Clip a deep-fry thermometer to the side of the pan and heat the oil to 375 degrees F.
Sift the flour, cocoa, baking soda, cinnamon and nutmeg into a medium bowl. Beat the egg, sugar and salt in a separate medium bowl until combined. Combine the milk and vinegar in a small bowl; the milk will curdle slightly.
Place the chocolate in a small bowl. Melt the butter in a small saucepan over medium heat until it browns and begins to smell nutty, 3 to 4 minutes. Immediately add it to the chocolate and stir until the chocolate is melted. Combine this with the milk mixture, along with the vanilla extract. Add the chocolate milk mixture in two additions to the eggs and sugar, alternating with the dry ingredients. Stir until a dough forms.
Turn the dough out onto a lightly floured a work surface. Roll out the dough out until it is 1/4 inch thick. Use a doughnut cutter to cut out doughnuts.
To test the oil, add a doughnut hole to the oil and see how it fries; it should float up to the top and have large bubbles surrounding it. When the oil is ready, fry the doughnuts and holes in batches, being careful not to overcrowd the pot. Fry until golden brown, about 3 minutes per side. They should be crisp on the outside and cooked through and fluffy on the inside. Drain the doughnuts on a plate covered with a paper towel. Cool each batch for at least 5 minutes and then coat with the ganache glaze. Serve the doughnuts the day they're made.