Make rosemary simple syrup: In a saucepan, combine water, sugar, and rosemary sprigs over medium heat. Stir until sugar dissolves (2-3 minutes). Remove from heat, steep for 10 minutes, strain, and cool.
Juice grapefruit: Squeeze fresh grapefruit to yield 1 oz juice, strain to remove pulp.
Shake cocktail: In a shaker, combine gin, grapefruit juice, rosemary simple syrup, and egg white (or aquafaba). Dry shake (no ice) for 15 seconds. Add ice and shake for 10-12 seconds.
Strain and top: Strain into a chilled highball glass with ice. Top with club soda, stirring gently.
Garnish: Add a rosemary sprig and grapefruit slice. Serve immediately.