In a large bowl, gently whisk together the whiskey and sweetened condensed milk until smooth. Add the instant espresso, chocolate syrup, and vanilla, whisking until the espresso dissolves. Finally, gently whisk in the heavy cream until smooth. Don't whisk vigorously, since the idea isn't to beat air into the cream.
Pour the mixture into a container with a tight fitting lid, and chill for at least 4 hours to allow the flavors to come together. The Irish cream will continue to develop and improve its flavor for the next two weeks, but it tastes wonderful and is ready to serve as soon as it's chilled. Serve plain over ice, in coffee, or in your favorite Irish cream cocktail.
Store in an airtight container in the refrigerator. If the cream is fresh, this should last at least 2 weeks if not longer since the alcohol acts as a preservative.