Trim about one inch off the end of the broccoli stalk, then peel away some of the stem near the cut end. It is tough and fibrous and won't puree nicely. Peel and smash the garlic with the flat side of a knife or chop it roughly. Peel and quarter the shallot or a small onion.
Melt the butter and olive oil.
Arrange the broccoli, shallot, and garlic on a sheet pan. Drizzle with the melted butter-oil mixture. Sprinkle with salt and pepper and roast in a 425-F oven for about 20 minutes.
Scrape all the broccoli, garlic, and shallot and any pan juices into a pot (save a few broccoli florets to garnish the soup.) Add 4 cups of low- or no-salt added chicken broth. Simmer over medium-high heat for about 15 minutes or until the broccoli is super-soft.
Using an immersion blender, puree the broccoli, garlic, shallot, and broth until smooth. Alternatively, you can puree it in a blender.
Add one cup of cream, and heat the soup to barely simmering. Adjust the salt and pepper, top with a bit of the reserved chopped broccoli florets, and serve!