Ingredients

• 1 3/4 cups milk
• 3/4 cup heavy (whipping) cream
• 2/3 cup water
• 1 cup sugar
• 1/3 cup light corn syrup
• 1/8 teaspoon kosher salt
• 1 1/2 pints (3 cups) fresh blackberries
• 3 tablespoons crème de cassis
• 1 teaspoon freshly squeezed lemon juice
• 2 small fresh pineapples
• 6 tablespoons sugar, plus more as needed
• Large pinch of kosher salt
• 3 cups cold seltzer water
• 2 cups ice cubes

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Instructions

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.
2. Purée the blackberries in a food processor. Strain the purée through a medium-mesh, then a fine-mesh sieve, discarding the seeds. There should be 3/4 cup pure. Stir the purée, 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.
3. To make the soda: 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. Purée 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.
4. 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. Purée 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.
5. To serve: Put the pineapple purée 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.
6. Put the pineapple purée 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.
7. In advance: The pineapple purée can be made a day ahead and refrigerated. Mix it with the seltzer just before serving.
8. The pineapple purée can be made a day ahead and refrigerated. Mix it with the seltzer just before serving.