How to Make It
Combine 1/2 cup water and sugar in a saucepan. Bring to a boil over high, stirring until sugar dissolves. Remove from heat. Add mint leaves and lavender. Steep 12 minutes. Pour through a fine wire-mesh strainer into a bowl; discard solids. Cool completely, about 10 minutes.
Combine cucumber and lime juice in a blender; process until smooth.
Stir together mint-lavender syrup, cucumber-lime mixture, and club soda in a large pitcher. Serve over ice; garnish with mint sprigs and lime wedges, if desired.