Put all the ingredients in a heavy bottomed sauce pan
Bring to a boil, then reduce the heat to a simmer. Cook until the juices are very thick and almost jammy.
Cool and then chill.
For the Ice Cream
Place the buttermilk, cream cheese and vanilla in a large bowl. Set a fine mesh strainer over the bowl and keep convenient to the stove.
Over medium heat, bring the half and half, sugars, salt and corn starch to a oil, whisking constantly.
Allow the mixture to boil for about 10 seconds, then strain it through a fine mesh strainer into the buttermilk mixture.
Let sit for a minute or two to melt the cream cheese and then whisk until smooth.
Chill in an ice bath, stirring occasionally, until cool.
Pulse 2 cups of the stewed peaches in a food processor until the peaches are fairly finely chopped.
Whisk the 2 cups of peaches into the dairy mixture. Press plastic wrap on the surface of the ice cream base and refrigerate until the temperature is no higher than 40F, at least 4 hours or preferably overnight.
Churn the ice cream in your churn according to manufacturer's instructions.
In a container large enough to hold all the ice cream, layer in about 1/3 of the remaining peaches and top with 1/3 of the ice cream. Continue layering until you've used all your ingredients.
Press plastic wrap directly on the surface of the ice cream and freeze until firm, 4-6 hours or overnight.