To slightly soften 5 cups vanilla ice cream, heat in a microwave oven at 30 percent power for 5 seconds, or let stand at room temperature about 15 minutes. Scoop ice cream into a large bowl.
Add 1/4 cup chopped crystallized ginger, 3/4 teaspoon ground cardamom, and 1/2 teaspoon ground cinnamon; stir quickly to swirl ingredients together. Place bowl in freezer until ice cream is firm enough to scoop, at least 1 1/2 hours; if storing up to 1 week, cover airtight.
Scoop the ice cream equally into 4 glasses (each 16 oz.). Slowly fill glasses with cold ginger beer or ginger ale, about 2 bottles (12 oz. each) total.