Bring pineapples, 2 cups water, and sugar to a boil in a large pot. Cover; simmer until fruit is tender, 5 minutes. Let cool slightly, then puree, in batches, in a blender. Strain into a bowl; let cool 1 hour. Use a muddler to crush mint in a large pitcher. Stir in pineapple juice; refrigerate overnight. Strain out mint, if desired. Add rum and lemon juice. Pour into ice-filled glasses, garnish with chopped mint and sprigs, and serve.