To make the raspberry simple syrup:
Combine the sugar and the raspberries in a small saucepan, stirring to blend together. Let them rest for about 30 minutes to allow the juices to mix with the sugar.
Add the water to the sugar and raspberries and warm over medium-low heat, stirring frequently until the sugar is completely dissolved, about 10 minutes. Take care not to boil the syrup. You're just warming enough to melt the sugar, not cook the raspberries.
Pour the raspberry simple syrup through a sieve to remove the seeds.
For the cocktails:
Pour the gin into a cocktail shaker.
Add the lemon juice and the raspberry simple syrup, followed by the vermouth and egg whites.
Cover the shaker and shake vigorously for about 10 to 15 seconds to create the foam.
Remove the top of the shaker, fill with ice and continue shaking for an additional 10 to 15 seconds or until well chilled.
Strain the cocktail between 2 coupes.
Garnish with a skewer of fresh raspberries.