Pour the ginger ale into a 9-by-13-inch baking dish. Freeze overnight.
Bring 4 cups water, the sugar and mint to a boil in a saucepan, stirring to dissolve the sugar. Let cool 2 minutes, then add the tea bags and let cool completely, 30 minutes to 1 hour; discard the tea bags.
Strain the sweet tea into a pitcher and add the bourbon. Refrigerate until ready to serve. Scrape the ginger ale ice with a fork and divide among glasses. Add the cocktail and garnish with mint sprigs.