Toast the cardamom pods in a saucepan over medium heat, stirring constantly, until fragrant, about 1 minute. Add 3 cups water, the sugar and ginger and bring to a boil. Reduce the heat and simmer until the mixture is reduced to 2 cups, about 10 minutes.
Strain the syrup into a glass measuring cup; if you have less than 2 cups syrup, add hot water. Add the tea bags and steep, about 3 minutes. Discard the bags and cool the syrup to room temperature (don't squeeze the bags or the syrup will taste tannic).
Combine the syrup, gin, vermouth and bitters in a large cocktail shaker with ice. Shake until frothy, about 15 seconds, then pour into a pitcher filled with ice. Garnish with orange and/or lemon twists.