Combine the apple juice, cardamom pods, allspice berries, cinnamon sticks, star anise, peppercorns, orange rind, and nutmeg in a saucepan over medium heat. Bring to a simmer, then reduce the heat to low and let it gently simmer, uncovered, for 20 to 30 minutes, until the juice has reduced by almost half and smells fragrant. Remove from the heat and let cool completely, then strain out the spices.

Pour the cooled, strained mulled apple juice into a pitcher. Add the apple slices, ice, and sparkling water or club soda, and stir gently to combine. Garnish serving glasses with rosemary sprigs and serve right away while it's still fizzy.