Combine the half-and-half and heavy cream in a medium saucepan and bring to a simmer over medium heat, stirring occasionally. Remove from the heat and set aside. Whisk the egg yolks in a large mixing bowl until they lighten in color. Gradually whisk the sugar into the yolks until smooth.
Slowly ladle one-third of the hot dairy into the yolk mixture, whisking constantly. Return this mixture to the pot containing the rest of the dairy. Cook over low heat, stirring frequently, until the custard thickens slightly, enough to coat the back of a spoon and reaches 170 to 175 degrees F.
Wash the original mixing bowl. When the custard is ready, transfer to the bowl, whisk in the coffee liqueur, espresso powder and vanilla, and cool at room temperature for 30 minutes. Cover and refrigerate until the temperature drops below 40 degrees F, 3 to 4 hours.
Liquid nitrogen procedure: Pour the ice cream base into the bowl of a stand mixer fitted with the paddle attachment. Turn the speed to low. With the mixer running, slowly pour in the liquid nitrogen while counting to ten. Increase the speed to medium and mix for 1 minute. Serve immediately or store in freezer.
Ice cream maker procedure: Pour into a prepped ice cream maker and process according to the manufacturer's directions. Within 25 to 30 minutes the ice cream will attain a classic soft-serve consistency. Serve as is for soft serve or freeze for another 3 to 4 hours to allow the ice cream to harden.