In a pitcher, stir together the lemon juice, Cointreau, cranberry juice cocktail, vodka, and egg white (if using). Fill a cocktail shaker half full with ice and pour enough of the drink mixture into the shaker to almost fill it. Shake the cocktail for a full 30 seconds (it's longer than you think!) and strain into martini glasses. Serve ice cold.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.