In a small saucepan, bring sugar, water, and 1/2 cup thyme sprigs to a boil over medium-high, stirring until sugar is dissolved. Remove from heat and let cool for about 30 minutes. Pour through a fine wire-mesh strainer into a bowl, and discard solids. Store thyme simple syrup in an airtight container in refrigerator up to 2 weeks.
Place lemon slices and 2 thyme sprigs in a cocktail shaker. Add carrot juice, Pimm's No. 1 liqueur, lemon juice, and 2 tablespoons of the thyme simple syrup. Muddle solids at bottom of shaker, and fill shaker with ice. Cover and shake vigorously until chilled, about 10 seconds. Strain mixture evenly into 2 ice-filled glasses. Top each with 1/4 cup ginger beer, and stir to blend. Garnish, if desired.