In a small saucepan over medium heat, combine sugar, water, and lavender. Allow to come to a boil, then reduce heat to low and allow mixture to simmer for 10 minutes.
Remove from heat and allow to cool slightly before draining through a sieve to remove lavender buds.
Fill a cocktail shaker with ice. Pour in 1/2 oz of the simple syrup, along with Absolut Elyx, coconut water, egg white, and lemon juice. Shake until frothy, then pour into a coupe glass. Top with edible flowers or more dried lavender.