Pour the water into a saucepan, and bring to a boil; stir in sugar until dissolved. Bring the syrup back to a boil, reduce heat, and boil gently until the syrup is reduced to 1 cup, about 10 minutes.
Remove from heat, and allow to cool.
In a mixing bowl, mash the avocados to your desired consistency, either smooth or chunky.
Mix in the evaporated milk, 1% milk, and lime juice. Stir in the cooled sugar syrup until completely incorporated. Chill the mixture completely in the refrigerator, about 4 hours.
Pour the chilled mix into an ice cream maker, and freeze according to manufacturer's instructions; pack the ice cream into a lidded container, and freeze until firm, at least 1 hour.
