Pass the scrubbed carrots through your juicer according to the manufacturer's instructions.
Pour the resulting juice through a fine-mesh strainer or cheesecloth into a clean bowl to remove any lingering pulp or foam.
Stir the fresh lemon juice and the grated ginger into the carrot liquid, ensuring the ginger is well-distributed for a subtle heat.
Place the bowl in the refrigerator for at least 10 minutes to ensure the juice is thoroughly chilled.
Pour 2 ounces of the prepared carrot mixture into each of the four chilled champagne flutes.
Slowly top each glass with approximately 4 ounces of cold sparkling wine, pouring at an angle to preserve the carbonation.
Stir very gently with a long cocktail spoon once to combine the ingredients without losing the bubbles.
Garnish each glass with a fresh mint sprig and serve immediately while cold.