In a small saucepan over medium heat, combine the sugar, water, rosemary, and cinnamon stick. Simmer the mixture until the sugar is dissolved and the rosemary and cinnamon start to smell fragrant, about 3 to 5 minutes. Remove the saucepan from the heat and let the rosemary and cinnamon steep in the liquid for about 15 minutes.
Add the chopped pears to your blender. Then strain the rosemary-cinnamon simple syrup directly into your blender, collecting and discarding the rosemary and cinnamon stick.
Blend on high until completely smooth. Transfer the pear purée to a clean container and refrigerate it until chilled.
For the bellinis, place about 2 tablespoons of the chilled pear purée into champagne flutes. Slowly fill with prosecco. Garnish with a sprig of fresh rosemary, if desired. Enjoy immediately.
This recipe yields about 2 cups of pear purée, which will be enough to make about 12-16 cocktails. One bottle of Prosecco will be good for about 5 bellinis, depending on the size of your champagne flutes.