Make basil syrup: In a large microwave-safe measuring cup, combine sugar, 3/4 cup water, and basil.
Microwave on high until sugar dissolves, 3 to 4 minutes.
Let basil steep in syrup 20 to 30 minutes, then strain (you'll have enough syrup for 4 drinks).
Fill 2 highball glasses with ice. Into each glass, pour 3 tbsp. gin, 4 tbsp. basil syrup, and 2 tbsp. lime juice. Stir and serve, garnished with basil sprigs.
Tip: Add lemon juice and chilled water to the leftover syrup and serve over ice for basil lemonade.