For the perfect layer of creamy froth on top of each cocktail, be prepared to shake the cocktail shaker (with no ice) vigorously; the half-and-half helps set the egg white for a more stable foam.
Place berries and sugar in a saucepan over medium. Cook, stirring occasionally to break up berries, until juices release completely, about 15 minutes. Pour through a fine sieve, pressing to release liquid; discard solids. (You will have about 1 cup liquid.) Cool mixture completely; stir in lemon juice.
Place half of blackberry syrup, 5 tablespoons gin, 1 egg white, and 1 tablespoon half-and-half in a cocktail shaker. Shake vigorously until frothy, about 15 seconds. Divide evenly between 2 glasses. Top each serving with 1/4 cup club soda. Repeat with remaining ingredients. Garnish with mint, if desired.