1. Make gastrique: Bring sugar and 1/4 cup water to a boil. Reduce heat to medium and cook until syrup starts to brown. Remove from heat. Slowly whisk vinegar into syrup. Add rum and figs; simmer 15 minutes and chill.
2. Make cocktail: Muddle 8 basil leaves, 2 strawberries, gastrique, and ice in a cocktail shaker, until fruit is in small chunks. Add rum, Cointreau, and bitters and shake. Strain into 2 ice-filled glasses and top with a splash of soda. Garnish each glass with 2 strawberry quarters and a basil leaf.