Combine sugar, water, and 17 mint sprigs in a small saucepan. Bring to a boil, stirring until sugar dissolves. Remove from heat. Strain and discard mint. Chill syrup several hours or overnight.
To serve, fill each julep cup three-fourths full with shaved ice; add 2 jiggers (1/4 cup plus 2 tablespoons) bourbon and 1 teaspoon reserved syrup. Stir gently. Add additional shaved ice to fill cups. Garnish with additional mint sprigs.