Make the pickle juice syrup: In a small saucepan, combine the pickle juice and sugar over high heat. Bring to a boil and cook until the sugar has dissolved, 1 minute. Remove from the heat and let cool completely.
Make the cocktail: In a shaker, combine the bourbon, lemon juice, syrup and egg white, and shake until frothy. Add ice and continue to shake until well chilled. Pour into a chilled coupe and garnish with a cornichon, then serve.