In a food processor, purée cabbage and 1/4 cup water. Strain and stir in baking soda; mixture will turn blue. Stir 1 tsp. cabbage liquid into the gin. Discard remaining cabbage liquid.
Make syrup: In a small covered saucepan, bring 1 cup water and remaining syrup ingredients to a boil; then reduce heat and simmer 20 minutes. Let cool; strain.
Make cocktail: Pour gin into a small glass. Combine liqueurs with remaining ingredients plus ó oz. syrup in another small glass. Set a large ice cube in a wide wine glass. Pour liqueur mixture on top, add gin, and watch the color change. Stir.
MAKE AHEAD The cardamom syrup, up to 1 month, chilled.
*Find gin at bottle shops and empress1908gin.com; rose water beverage at grocery stores and drinkh2rose.com; and rose water at grocery stores.