For the simple syrup: In a small saucepan, combine the thyme sprigs, sugar and 1/2 cup water. Bring to a boil over medium heat. Reduce the heat and simmer, stirring occasionally, until the sugar has dissolved, about 5 minutes. Remove the pan from the heat and allow the syrup to cool for at least 20 minutes.

Cut the ends off one of the grapefruits and place cut-side down on the counter. Using a paring knife, cut away the peel. Working over a small bowl or a large glass measuring cup to catch any juices, cut along the membrane on both sides of each segment and remove the segments. Place the grapefruit segments in 4 chilled champagne flutes. Squeeze the juice from the membranes into the bowl. Discard the membranes. Halve the remaining grapefruit and squeeze the juice into the bowl.

Strain the simple syrup and grapefruit juice into a small pitcher, pressing on the solids to extract as much liquid as possible. Add the prosecco.

Pour into the prepared glasses and garnish with fresh thyme sprigs if using