In a medium saucepan, combine the chopped carrots and water. Bring to a boil over medium heat, then reduce the heat and simmer for about 10 minutes, until the carrots are tender.
Drain any remaining water and allow the carrots to cool slightly.
In a blender or food processor, puree the cooked carrots with sugar, cinnamon, ginger, nutmeg, vanilla extract, and lemon juice until smooth.
Transfer the mixture to a mixing bowl. If using an ice cream maker, process according to the manufacturer's instructions until it reaches a soft-serve consistency. Otherwise, pour the mixture into a freezer-safe container.
Freeze the sorbet for at least 4 hours, or until firm.
To serve, scoop the sorbet into bowls and swirl in the mixed fruit jam. Garnish with fresh mint leaves if desired.
