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1
In a bowl, whisk together the milk, cream, water, sugar, corn syrup, and salt in a medium heavy saucepan. Cook over medium heat, stirring occasionally, until almost simmering. Transfer the mixture to a bowl and cool over an ice bath until room temperature.
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2
Puree the blackberries in a food processor. Strain the puree through a medium-mesh, then a fine-mesh sieve, discarding the seeds. There should be 3/4 cup pure. Stir the puree, cassis, and lemon juice into the cream. Refrigerate for 4 hours, or up to overnight. Churn in an ice cream machine according to the manufacturer's instructions. Freeze the sherbet until scoopable, about 4 hours, depending on your freezer.
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3
Remove the rind from the pineapples. From the top, cut the pineapples into quarters. Remove and discard the core. Cut the pineapple flesh into 1-inch pieces. Puree in a food processor until smooth. Strain through a medium-mesh sieve, discarding the pulp. There should be about 4 cups pure. Refrigerate the pure for at least 30 minutes or up to 24 hours.
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4
Put the pineapple puree in a large pitcher. Stir in the 6 tablespoons sugar and the salt. Taste for sweetness, adding a little more sugar if necessary. Stir in the seltzer water. Grind the ice in a food processor until it has the consistency of shaved ice. Divide the ice among eight 12-ounce glasses. Fill with pineapple soda. Put several scoops of cassis-berry sherbet in each glass. Serve immediately, with straws and long spoons.
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5
The pineapple puree can be made a day ahead and refrigerated. Mix it with the seltzer just before serving.
