Melt butter in 3-quart saucepan over medium heat.
Add onion and saute until soft.
Add stock and garlic and bring to simmer.
Cover and cook 30 minutes.
Let mixture cool slightly.
Puree in batches in processor or blender.
Transfer to large serving bowl.
Puree avocado and lime juice and add to stock.
Blend in buttermilk and season with salt and pepper to taste.