In a medium saucepan, bring 4 cups water to a boil. Stir in the sugar and cook for about 5 minutes. Add the tea bags and mint, then remove from the heat and steep for 10 minutes. Strain the mixture and discard the tea bags and mint. Cool for about 1 hour, and then stir in the lemon juice.
Fill 8 smallish glasses (or classic julep cups) with crushed ice and muddle in a few leaves basil and mint in each glass, bruising the herbs and mixing them with the ice. Add a squeeze from the limes to each glass, then pour about 3 1/2 ounces of the cooled syrup and about 2 1/2 ounces of the bourbon per glass. Add a straw and enjoy.