Whisk eggs and molasses together in a medium bowl until combined. Set aside.
Combine cream, buttermilk, brown sugar, vanilla extract, salt, and cinnamon in a medium saucepan over medium heat. Cook, stirring frequently, until sugar is dissolved and mixture begins to bubble around the edges, about 5 minutes.
Add 1/2 of the hot cream mixture very slowly into the egg-molasses mixture, whisking constantly. Pour the tempered egg mixture back into the pan with the remaining hot cream mixture. Reduce heat to medium-low and continue to cook until mixture is slightly thickened and coats the back of a spoon, about 8 minutes.
Strain custard through a fine-mesh sieve into a medium bowl. Cover with plastic wrap touching the top of the custard to avoid forming a skin. Cool completely in refrigerator, 4 hours to overnight.
Pour custard into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.