For the pear simple syrup:

1. In a small saucepan, add water, sugar, pear, and rosemary over medium heat, and bring to a light boil, stirring occasionally, until sugar is dissolved.
2. Remove from heat and let sit until cools to room temperature (about 60 minutes).
3. Strain syrup, discarding the rosemary and pear, or saving the pear for a topping for oatmeal, yogurt, pancakes, etc.
4. Transfer the simple syrup into a glass jar and store in the refrigerator for up to 3 weeks.  

For the cocktail:

1. In a cocktail shaker, add gin, lemon juice, pear simple syrup, and egg white (or aquafaba) and shake vigorously for about 15 seconds.
2. Add ice to cocktail shaker and shake again until cold, about 30-60 seconds.
3. Strain drink into glass and fill to the top with soda water.
4. Optional: Garnish with a pear slice and sprig of rosemary.

Notes:

1. Consuming raw or undercooked eggs may increase your risk of foodborne illness.
2. Aquafaba is the liquid leftover from canned chickpeas and is a vegan substitute for the egg white.