How to Make It
In a large saucepan, combine 5 cups of the cranberries with the water and 1 3/4 cups of the sugar and bring to a boil, stirring to dissolve the sugar. Cook over moderate heat until the cranberries split, 4 to 5 minutes. Pour the cranberries into a colander set over a bowl without pressing on the solids; reserve the cranberry syrup.
In a blender, chop the cooked cranberries with 1/2 cup of the sugar. With the machine on, pour in 1 cup of the reserved cranberry syrup and puree until smooth. Strain the cranberry sauce in a fine sieve and refrigerate until chilled.
Combine the heavy cream with the cold cranberry sauce and freeze in an ice cream maker according to the manufacturer's instructions. Transfer the ice cream to an airtight container and freeze until firm, at least 2 hours.
In a saucepan, combine the remaining 1 1/2 cups of sugar with 3 tablespoons of the cranberry syrup and stir until the sugar is moist and pink. Cook the sugar over moderate heat, brushing down the sides of the pan with a wet brush, until a light amber caramel forms, about 8 minutes. Remove from the heat and add 1/2 cup of the cranberry syrup, gently swirling until blended.
Add the remaining 3 cups of cranberries to the caramel sauce and cook over moderate heat, gently pressing the cranberries against the sides of the pan, until slightly softened, about 5 minutes. Stir in the raisins and currants and transfer the compote to a large bowl.
In a saucepan, combine the remaining cranberry syrup with the vanilla bean and cinnamon stick and bring to a boil. Add the pears, cover and cook over moderately high heat, stirring, until tender, 10 to 12 minutes. Let cool slightly, then strain the pears. Discard the vanilla and cinnamon. Stir the pears into the compote. Reserve 1 cup of the pear cooking liquid for serving. (You can reserve the remaining cooking liquid for poaching fruit at a later time.)
In a food processor, combine the flour, sugar and salt and pulse to blend. Add the butter and pulse until the mixture is the size of peas. Add the ice water and process just until the dough comes together. Transfer the dough to a lightly floured work surface and knead it several times. Shape the dough into a log that is 3 inches wide. Wrap the log in plastic and refrigerate until chilled, at least 1 hour.
Unwrap the dough and cut it into 12 rounds. Working with 1 at a time, roll each piece of dough out to a 6-inch round. Transfer the rounds to 4-by-1-inch fluted tartlet pans with removable bottoms, pressing the dough into the corners. Trim the edges and prick the bottoms several times with a fork. Refrigerate the tartlet shells until chilled.
Preheat the oven to 400°. Line the tartlet shells with foil and fill the foil with pie weights or dried beans. Bake for about 10 minutes. Carefully remove the tartlet shells from the oven and remove the foil and weights. Bake the tartlet shells for about 20 minutes longer or until golden. Let the shells cool completely on a wire rack.
Unmold the tartlet shells and set them on small plates. Fill the shells with the fruit compote. Spoon a scoop of the cranberry ice cream alongside each tartlet and spoon the reserved poaching liquid over top. Garnish the tartlets with mint sprigs and serve.