In the jar of a blender, place 1 cup milk, muscavado sugar, molasses, pumpkin puree, cinnamon, ginger and nutmeg. Blend on high until smooth, about 30 seconds. Transfer to a bowl, whisk in remaining milk and heavy cream. If not making ice cream immediately, cover and refrigerate until ready to use.
Turn on ice cream maker. Pour pumpkin mixture into freezer bowl. Mix until desired consistency, about 20 to 25 minutes. Begin checking consistency after 12 to 15 minutes. Serve in cones or cups, adding mix-ins while dispensing.