Combine cherries, 1/2 cup sugar, lemon juice, and balsamic vinegar in a bowl and mix well. Set aside for 2 hours. Drain, reserving juice.
Mash 1/2 the cherries with a fork or puree in a blender.
Combine milk and remaining 1/2 cup sugar in a bowl and beat with an electric mixer until sugar has dissolved, 1 to 2 minutes. Add reserved cherry juice, cream, pureed cherries, vanilla extract; mix to combine.
Pour mixture into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Add remaining quartered cherries and freeze for an additional 5 minutes. Ice cream should be nice and creamy. If you prefer it harder, transfer to an airtight container and freeze until firm, about 4 hours. Remove from freezer 10 minutes before serving.
