In a blender, combine the baby cut carrots, sliced ginger, and water. Blend until smooth. Strain the mixture through a fine mesh strainer to reserve 1 cup of juice.
Fill 4 tall glasses with ice cubes. In a cocktail shaker, combine 2 tablespoons lemon juice, ¼ cup carrot-ginger juice, and 1 tablespoon simple syrup per glass. Add ice and shake well. Strain the mixture over ice in the prepared glasses. Top each with club soda.
Garnish each drink with fresh carrot top leaves and a thin carrot slice, if desired. Serve immediately.