1. In two steps prep the broccoli, peel toughest part of stem then dice into cubes. Cut off then dice the stem off the broccoli florets. Trim the broccoli florets to small pieces, set aside in bowl.
2. Boil the diced broccoli stems with a little salt, eiher use 2 cups of water or just add enough water to cover the stems and cook until semi tender then add the broccoli florets with 1 teapoon of Better Than Broth for each 1 cup (8 oz) of liquid added. When the broccoli florets are just tender, use a hand blender to blend out the broccoli. **Remove 1 or so cups of the florets, not the stem and add after blending if some whole florets are preferred in your soup.
3.(2 options): Either boil the grated carrot and add or add after blending the soup if you prefer the texture of the grated carrot in the soup.
4. Saute diced onion in olive oil until translucent, add the minced garlic, attentively saute' about 3 minutes stirring making sure the garlic does not burn. Add to the soup pot.
5. Add 1 Tablespoon of dried Tarragon.
6. The Roux: Melt butter in saute' pan with flour, making roux stirring a few minutes until golden brown, slowly add the half and half or milk until thickens adding additional half and half or milk as needed for consisitency, add 1/2 to 1 cup grated cheddar cheese stirring until cheese has melted. Then ladle some of the soup into the cheese sauce stirring, tempering adding several ladles of soup, each time stirring. Once tempered add the sauce to the soup pot and stir well.
**If you prefer to stay with the classc roux of equal butter, flour and liquid that is an option. However, I use a cube or 1/2 cup of butter when I double the recipe.
7. Add 1/4 cup grated parmesan cheese.
I tend to take it easy on the salt when adding cheese, but I do add a little salt at each step.
*Making adjustments: to thicknen: cut up more broccoli stems boiling until tender in salt water (in a separate pan), then hand blending and adding, stirring into the soup. This soup gets devoured in our home, so I double this recipe for my family.