Steps

1. Heat 1 cup of the heavy cream and ¼ cup of the rum in a small pot over medium heat until almost boiling. Set aside.
2. To make the caramel: Place 1 cup of the sugar in a heavy-bottomed skillet over medium-high heat. Cook until sugar begins to melt; when you see large patches of melted sugar, stir the sugar and continue to cook until sugar is completely melted and amber in color. Slowly add the cup of warm cream (with rum) and stir to combine. Transfer to a shallow bowl and refrigerate.
3. Place 1 cup of the peaches in a microwave-safe bowl and add the remaining 2 tablespoons rum. Cover with plastic wrap and microwave 1 to 2 minutes, or until peaches are warm and rum is absorbed. (If you don’t have a microwave, place peaches and rum in a small saucepan over medium-high heat. Cover and cook 2 to 3 minutes, or until peaches are heated through and almost all of the liquid is absorbed.) Set aside to cool. When cool, chop the peaches into small pieces.
4. Meanwhile, place remaining peaches and remaining 1½ cups sugar in the bowl of a food processor and process until smooth. Pass mixture through a fine mesh strainer, discarding solids. Place strained puree in a large bowl, add whole milk and remaining 2 cups cream, and whisk until combined.
5. Pour mixture into ice-cream maker and process according to your ice-cream maker’s instructions. When mixture is the consistency of soft-serve ice cream, stir in reserved chopped peaches and caramel. Transfer to a freezer-safe container and freeze until firm, at least 3 hours.