Make syrup: Bring 2 cups water and sugar to a boil over medium-high heat in a medium saucepan, stirring until sugar is dissolved. Add tarragon leaves, reduce heat to medium, and simmer 2 minutes. Remove from heat, and let stand to infuse for 1 hour.
Strain liquid through a fine-mesh sieve, squeezing out as much liquid as possible from tarragon leaves. Refrigerate in an airtight container until ready to serve, up to 1 week.
To serve, combine 1 ounce tarragon syrup and 1/2 ounce vodka in a Champagne flute. Top with Prosecco (about 5 ounces), and garnish with tarragon sprigs.