Add the raspberries and sugar to one medium pot and the peaches and sugar to another. Cook both on medium low heat until the sugar is melted and the fruit breaks down into a sauce.
Cool the mixtures slightly and then strain them. Push some of the pulp through the strainer, but don't get too many raspberry seeds in the raspberry mixture. Set the juice aside to cool completely
MAKE THE COCKTAIL
Add the rum, lime, and mint to a jug that you can reach into. Muddle them well with a muddler or something that you can really smush the mint with to release the flavor.
Add the fruit juices and the bottle of prosecco. Pour into icy glasses and serve with a sprig of mint to make it look pretty!