In a stock pot, melt 2 TBS butter. Once melted, add in chopped onion and cook until becomes translucent. About 3-5 minutes.
Add chopped broccoli florets, cook and stir about 5-6 minutes.
Pour in 6 cups of chicken broth and allow for the this to come to a rapid boil. In about 4 minutes, poke a piece of broccoli with fork. If it easily goes through, it’s finished.
While the broccoli is boiling, make your thickening agent in another pan. I used a small skillet to combine 4 TBS unsalted butter and 4 TBS all-purpose flour with whisk. Allow for the flour taste to cook out, about 1-2 min on low-medium heat.
With an immersion blender, blend soup to desired consistency. I left some smaller pieces for texture.
Bring to a low-medium boil and whisk roux (thickening agent) into your broccoli. Allow to thicken while stirring.
Add salt and pepper and stir.
Stir in half-and-half and stir. The soup will continue to thicken as it sits.
Garnish with cheese or eat how it is. Both taste amazingly good.