Put lavender blossoms, sugar, and 1/2 cup water in a microwave-safe bowl. Microwave until very warm, then stir to dissolve sugar. Let sit at least 20 minutes but no more than 1 hour, then strain syrup.
Pour 1/4 cup syrup, carrot juice, and vodka into a cocktail shaker filled with ice. Shake well to blend, then strain into 2 martini glasses. Garnish glasses with lavender sprigs. Chill remaining syrup to use another time.
Note: Nutritional analysis is per serving.