To make the sauces: in medium saucepan, combine the berries (do the strawberries and blueberries in separate pans, or do one berry then wash the pan and do the second berry) and the sugar. Set over medium-high heat and cook, stirring constantly, until the berries begin to soften. Use a potato masher to mash the berries until any large lumps are gone. Cook, stirring constantly, until the sauce thickens, 5-10 minutes. Pour into a Mason jar or lidded bowl and refrigerate until completely cold.
To make the ice cream: using an electric mixer, whip the cream until stiff peaks form. Pour the sweetened condensed milk into a large mixing bowl and stir in the vanilla extract. Fold in about 1/3 of the whipped cream, then repeat with thirds until all of the cream has been mixed in.
Layer the ice cream mixture into a loaf pan, large storage container, or ice cream containers, alternating thin layers of the ice cream mixture with spoonfuls of the fruit sauce, alternating sauces. Freeze until hard, at least 4 hours. Serve with remaining sauce and/or fresh strawberries and blueberries.