Stir the cider, cinnamon sticks, ginger, cloves and nutmeg together in a pot over medium heat. Bring to a boil, then reduce the heat to low and simmer until reduced by half, about 30 minutes. Set aside to cool, then strain.
In a large bowl, toss the apples, lemon juice and sugar. Add to the cider and refrigerate until chilled. When ready to serve the cocktails, fill champagne flutes about halfway with the cider mixture and top with the cava. Stir and serve.