Heat cream in a small saucepan until small bubbles appear around the edges. Remove from the heat and stir in sugar until it dissolves. Transfer mixture to a medium bowl and allow to cool, about 10 minutes.

Blend avocado and cream cheese in a food processor until smooth. Add lime juice and process until smooth.

Whisk blended avocado mixture into cooled cream mixture, then pour into an ice cream maker. Freeze according to the manufacturer's instructions, about 20 minutes.

Spoon avocado mixture into the prepared pie crust; smooth the surface with a spatula. Freeze until solid, at least 2 hours. Let cheesecake sit at room temperature until slightly softened yet firm enough to slice before serving, 10 to 15 minutes.