Combine the cream, milk, sugar, and coffee in a medium, heavy saucepan. Bring to a gentle boil over medium heat. Remove from the heat.
Beat the egg yolks in a medium bowl. Whisk 1 cup of the hot cream into the egg yolks. Gradually add the egg mixture in a slow, steady stream, to the hot cream. Cook over medium-low heat, stirring occasionally, until the mixture thickens enough to coat the back of a spoon and reaches 170 degrees F. on an instant-read thermometer, about 5 minutes. Remove from the heat and strain through a fine mesh strainer into a clean container. Cover with plastic wrap, pressing down against the surface to keep a skin from forming. Chill in the refrigerator for 2 hours.
Remove from the refrigerator and pour into the bowl of an ice cream machine. Freeze according to the manufacturer's instructions. After the ice cream is made, transfer to an airtight container. Cover tightly and freeze until ready to serve. Garnish with chocolate curls.