Fill a cocktail shaker (or a mason jar) with the apple juice, lemon juice, grenadine, and egg white. Place the lid on (or screw a mason jar lid on tight) and shake vigorously for 30 seconds. Add ice and shake for another 30 seconds or until chilled.
Add the club soda to a glass then strain the mocktail mixture into the glass. Garnish with maraschino cherries and serve.
Notes
To make a large batch in advance, mix together the first four ingredients - grenadine syrup, apple juice, lemon juice, and egg whites - and keep in the fridge for 3-4 days. When ready to serve, shake the ingredients to emulsify, then pour over club soda and garnish with cherries. If you can't find grenadine syrup, the syrup from the maraschino cherries will do more in a pinch. Leftovers can be stored in the fridge for 1-2 days but will lose dizziness.