Fill the shaker with ice. Combine the vodka, apple juice, and thyme syrup in a martini shaker. Shake for about 10 seconds. Divide between the 4 chilled martini glasses.
Use the melon baller to scoop out small balls of apple. Place 3 balls of apple and 1 sprig of thyme in each glass for garnish. Serve immediately.
Thyme Simple Syrup:
In a saucepan combine sugar, water, and thyme over medium heat. Bring to a boil, reduce heat and simmer until the sugar has dissolved, about 5 minutes. Take the pan off the heat and cool the syrup. Any extra cooled syrup can be saved in an airtight container in the refrigerator.