Halve the kumquats and remove any large, onerous seeds. Set aside in a large bowl.
Prepare a simple syrup, enough to cover the fruit. Add the sugar and water to a large pot, bring to a boil, and then reduce to a simmer until the sugar has completely dissolved.
Transfer the halved kumquats the the pot and bring to a boil once again. Reduce the heat and simmer for about 20 minutes, or until the fruit is soft and candied. Your kitchen should smell absolutely glorious.
Let cool completely. Pour the roasted kumquat syrup into a blender, puree, and ladle into canning jars.
Combine equal parts kumquat syrup and bourbon in a drink mixer. Add ice and the juice of half a lemon. Shake until mixed. Pour into glasses until they're 3/4 the way full.
Top each drink with prosecco and serve. Add a sprig of mint or an extra rind as a garnish. You'll have extra kumquat syrup; store in the freezer until a hot summer day, and pull it out to impress your friends at a park picnic.