3–5 tablespoons of the blackberry-thyme syrup (or to taste)
¼ cup cold seltzer water
Fresh blackberries, for garnish
For the Syrup:
Combine the sugar and water in a small saucepan and bring to a boil, stirring to dissolve the sugar. Simmer for a couple minutes and then remove from heat. Add the thyme and berries and allow to steep until cooled to room temperature. Remove the thyme and then pour the syrup and berries into a blender. Blend until smooth. Strain through a fine-mesh strainer to remove the seeds. Keep cold in an airtight container until ready to use.
For the Cocktails:
Place the gin, lemon juice, lime juice, egg white, half-and-half, and blackberry-thyme syrup into a large shaker (without ice). Shake vigorously for about a minute, then fill the shaker with ice, and continue to shake until everything is icy cold and frothy. Strain the mixture into two glasses. Pour the seltzer water into the shaker to pick up the foam leftover from the shaking. Spoon the resulting fizz on top of the glasses and garnish with fresh berries. Serve immediately.