Whisk the heavy cream, milk, brown sugar and granulated sugar together in a medium sauce pot then bring it to a simmer over medium-low heat while stirring occasionally.
Once the mixture is full cooled, pour it into the ice cream maker bowl and begin churning according to the manufacturer’s instructions. Mine takes about 20 minutes. Add in the toasted marshmallow mixture at the very end and allow to churn 1 more minute.
Once it’s churned it will be soft serve consistency. You can eat it just like this or pour it into a loaf pan then freeze it for 6-8 hours or overnight.