In a high-speed blender, combine the chopped avocado, 1. 5 cups (1200 ml) of the milk, a generous pinch of salt, and a small pinch of cayenne pepper.
Blend until completely smooth and creamy.
With the blender on low, stream in the remaining 1. 5 cups (1200 ml) of milk until fully incorporated. For the best flavor and texture, transfer the soup to a container and chill for at least 30 minutes.
Before serving, taste and adjust the seasoning with more salt or cayenne if needed.
Stir in the fresh orange or lime juice, if using.
Pour into chilled bowls and serve immediately.