Heat a medium sauce pan over medium heat for 4 - 5 minutes.
When the pan is hot, add the butter and oil.
Once the butter is melted, add the onion and celery and saute until the vegetables are translucent and just starting to turn golden.
Stir in the additional butter. When melted, stir in the flour. Cook, stirring constantly. for two-three minutes.
Pour in the chicken broth, broccoli stems and half and half along with another heavy pinch of salt. Simmer until the broccoli stems are very soft, about 10-12 minutes.
Blend with an immersion blender or in batches in a conventional blender until the soup is as smooth as you like it.
Taste and adjust seasonings.
Serve as is with some grated cheese or use in any recipe that calls for cream of broccoli soup.