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Note: Prep time does not include time for infusion and chilling.
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Combine 3/4 cup of sugar into the bowl of a food processor. Zest all lemons directly into the bowl. Blend until the mixture is well combined and looks rather sandy.
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Combine 1/2 cup of cream, all of the milk, the lemon-scented sugar, and salt in a medium saucepan. Gently warm over medium heat, stirring, until all of the sugar is dissolved. Remove from heat, cover and allow the mixture to sit and infuse for an hour.
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Add the remaining 2 cups of cream and vanilla to the mixture and refrigerate for a few hours until completely chilled.
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Meanwhile, place the blackberries and remaining 1/4 cup sugar in a medium saucepan. Cook over medium heat, stirring frequently, until berries have cooked down, about 10-15 minutes. Strain the mixture, pressing to extract all of the liquid, and discard the seeds. Return the strained sauce to the pan. Continue to cook over medium heat until mixture has reduced about a third. At this point you can whisk in the cornstarch if you want your syrup to be extra thick. Refrigerate until ready to use. (By the way, this stuff would be fantastic over pancakes.)
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Freeze ice cream mixture according to the manufacturer's instructions. As you remove it from the machine, swirl in the blackberry syrup. I layered mine and it seemed to work well.
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Variation: If you use Meyer lemons or oranges, it will give the ice cream an orange dreamsicle flavor.
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(Source: Ice cream base adapted from David Lebovitz.)
