Add blackberries, water, and sugar to a small pan and bring to a boil over medium heat. Continue boiling for 5 minutes or until blackberries can be easily mashed with the back of a spoon. Mash the blackberries until they are all burst.
Turn off heat and add thyme sprigs. Allow to steep for 3-5 minutes, then strain out liquid with a fine mesh.
Divide blackberry syrup among 3 coupe glasses. Top with prosecco and garnish with additional blackberries and thyme, if desired.