Whisk together molasses, cinnamon, allspice, nutmeg, and ginger and set aside.
Heat heavy cream and milk in a medium saucepan over medium heat, stirring often, until it’s very hot – but do not let it boil. While it’s heating, whisk (or beat using stand mixer with wire whip attachment) the egg yolks and sugar together.
Add ¼ cup of the milk and cream mixture to the egg mixture gradually, whisking constantly. Continuing to whisk, gradually add the remaining milk and cream mixture. (Adding gradually and whisking constantly are really important here. Otherwise you’ll end up with yucky cooked eggs instead of an ice cream base.)
Return the mixture to the saucepan and heat over medium-low heat and cook until it reaches 170°. At this point, the mixture should have thickened and should be able to coat the back of a wooden spoon.
Remove the ice cream base from the heat and strain through a mesh strainer.
Whisk in the gingerbread flavoring mixture.
Cover and let it come to room temperature, then place in the refrigerator and chill for 8+ hours.