Place the avocado, lime juice, and sugar into a food processor. Cover, and puree until smooth; set aside. Beat whipping cream in a large glass or metal mixing bowl until medium peaks form. Lift your beater or whisk straight up: the tip of the peak formed by the whipped cream should curl over slightly.
Beat the eggs lightly in a saucepan, stir in the evaporated milk, condensed milk, and 1 cup whole milk. Heat over medium heat, stirring occasionally, until simmering. Continue simmering until very hot, 5 to 7 minutes. Remove from heat and cool slightly. Stir in the avocado mixture and the remaining 2 cups milk. Use a rubber spatula or wire whisk to fold 1/3 of the whipped cream into the avocado mixture. Gently run the spatula through the center of the bowl, then around the sides of the bowl, repeating until fully incorporated. Add the remaining whipped cream, folding just until incorporated. Cover and refrigerate until cold.
Pour the chilled mixture into an ice cream maker and freeze according to manufacturer's directions until it reaches "soft-serve" consistency. Transfer ice cream to a one- or two-quart lidded plastic container; cover surface with plastic wrap and seal. For best results, ice cream should ripen in the freezer for at least 1 hours or overnight.
