Make the lavender simple syrup by combining water, sugar, and dried lavender in a small saucepan. Bring to a boil, then reduce heat and simmer for 3 minutes. Remove from heat, let steep for 10 minutes, then strain out the lavender.
If desired, rim two martini glasses with sugar by running a lemon wedge around the rim and dipping in sugar.
Fill a cocktail shaker with ice, then add vodka, fresh lemon juice, triple sec, and 1 ounce of the lavender simple syrup.
Shake vigorously for 15-20 seconds until well chilled, then strain into the prepared glasses and garnish with fresh lavender sprigs and lemon slices.