In a blender, puree the mango and prosecco until very smooth.
Set a large metal bowl on a damp dish towel and fill it halfway with ice, water, and a cup of rock or kosher salt. The towel will help keep it from sliding around on the counter as you mix the sorbet.
Set a second, smaller metal bowl in the ice bath (make sure none of the water can spill into the upper bowl, drain some of water from the larger bowl if necessary).
Combine the mango puree, sugar, lemon juice, and kumquat slices in the smaller bowl and stir vigorously with a rubber spatula, constantly scraping the bottom of the bowl, until the sugar is dissolved. Continue to mix until very cold.
At this point, if you have an ice cream/sorbet maker, you can pour the mixture directly into the prepared mixing chamber and turn it on.
If you don't have an ice cream maker, you can continue to mix the sorbet in the bowl. Alternate placing the smaller bowl with the sorbet in the freezer for 10 minutes, then returning it the ice bath and mixing it for 5 minutes, until it sets. If necessary, drain the water and add more ice/salt to keep the bath super cold. You may have to switch to a harder plastic or wooden spoon to scrape up the sorbet as it freezes to the bottom of the bowl.
Once set, pack into a quart sized container and store in the freezer for several hours to firm up before serving.