Heat 2 tablespoons of the olive oil in a medium skillet over medium heat. Add the onion, chile and the garlic and cook until slightly soft, about 2 minutes.
Season with salt, to taste. Remove from the heat and set aside to cool.
Put the avocados in a large bowl. Add the chicken broth, lemon juice, cilantro, onion mixture, and water. Add, in batches, to a blender and puree until smooth, straining each batch of puree into a large bowl. Stir in the 1 teaspoon of salt and the 1 teaspoon of pepper, then cover and refrigerate until well chilled, about 3 hours.
Pour the chilled soup into individual bowls. Top each serving with a drizzle of the sour cream. Serve with crostini or croutons, if desired.
*Mexican sour crema is called cream or crema acida and can be found in Latin markets.