In a saucepan, whisk together heavy cream, whole milk, dark brown sugar, and molasses. Add fresh ginger pieces. Heat over medium heat and bring to a simmer, whisking occasionally.
In a separate bowl, whisk egg yolks, vanilla, cinnamon, allspice, cloves, and salt for about three minutes. Gradually add ¼ to ½ cup warm cream mixture into egg yolk mixture, whisking to temper eggs. Repeat until combined.
Pour tempered egg and cream mixture back into saucepan with remaining cream mixture. Cook over medium heat, whisking constantly, and bring to a simmer without boiling. Cook until thickened, about six to ten minutes. Remove from heat and stir in orange zest. Let steep for 10 minutes, stirring occasionally.
Strain custard through a fine mesh sieve into a large bowl. Press plastic wrap on surface to prevent skin. Refrigerate until cold, at least four hours or overnight.
When ready, pour chilled custard into ice cream maker and churn per manufacturer’s instructions. Add crushed ginger snaps in the last few minutes of churning.
Transfer ice cream to airtight container and freeze for a few hours before serving.