Heat peaches and sugar in sauce pan. Bring to a boil then reduce heat to low and allow peaches to simmer 2-3 minute. Remove from heat.
Pour into a blender and blend until smooth. Allow peach puree to cool to room temperature.
While peaches are cooling, heat oven to 350 degrees. Crush graham crackers and mix in melted butter and sugar. Spread in an even layer on a baking sheet and cook 6 minutes, until graham cracker crumbs are toasted. Let crumbs cool.
Mix heavy whipping cream, milk, vanilla and cinnamon in a bowl. Stir in peach puree.
Pour mixture into ice cream machine and process according to manufacturer instructions.
Stir in extra diced peaches towards the end of the churning process, if desired.
Scoop mixture into a freezer-safe container. Sprinkle top with graham cracker crumbs.
Enjoy right away for soft serve ice cream or freeze 3-4 hours for scoop-able ice cream.