Set a medium (3-quart) saucepan over medium high and add the sugar with 1 ¼ cups of water. Bring to a boil, whisking until the sugar is dissolved. Remove from the heat and cool completely.
Cut the avocados in half, running a knife around the pit from top to bottom and back up again. Twist the halves in opposite directions to release the pit from one side. Remove the pit, then scoop the flesh from each half into a food processor or blender and process until the avocado is completely smooth. You will need 3 cups of avocado puree.
Add the cooled syrup, lime juice and salt to the avocado and process until everything is well mixed.
Pour the mixture into the canister of your ice cream maker and freeze according to the manufacturer’s directions. The ice will have the best texture if scooped into a container and placed in a freezer to firm up for 3 or 4 hours.