Melt the butter in a large stock pot over medium heat. Add onion and garlic and sautée gently until softened and translucent, about 3 minutes.
Add chicken broth, broccoli and salt and bring to a boil. Reduce heat, cover and simmer for 13 minutes.
Allow the soup to cool a bit before pureeing it. Working in batches, place the 1/4 of the avocado and 1/4 of the soup into a blender and puree until very very smooth. Repeat with remaining avocado and soup.
Place soup back into the pot and whisk in heavy cream. Add more salt and pepper to taste. Do not boil. Whisk in sour cream and stir until well combined.
Serve with a broccoli flowerette or a tablespoon of shredded extra sharp cheddar cheese or a dollop of sour cream.