Heat 1/2 cup each sugar and water in a saucepan over medium heat until the sugar is dissolved. Add 1/2 stemmed jalapeño (remove seeds for less heat) and 2 sprigs thyme and let infuse, 5 minutes. Discard the jalapeño and set aside the thyme. Puree the simple syrup with 5 cups chopped seedless watermelon in a blender. Serve over ice and garnish with pieces of the reserved thyme.