Step 1 Cut the rind off the pineapple, then cut the fruit crosswise into 4 pieces.
Step 2 In a large saucepan, heat the water, sugar, vanilla bean and seeds, and cinnamon stick over medium heat. When the mixture comes to a boil, add the pineapple and simmer for about 3 minutes.
Step 3Remove the pineapple from the saucepan and let cool. Puree the pineapple in a food processor. Strain the puree and the contents of the saucepan through a fine-mesh sieve. Return the strained pineapple mixture to the saucepan over medium heat and boil until reduced by about two-thirds, 15 to 20 minutes. Cool.
Step 4In a medium bowl, combine the cooled pineapple mixture and the sake. Add the lemon juice. Pour into a glass container and refrigerate. Serve well chilled.