Saute onion and garlic until just soft but not browned. Chop half of the carrot into half inch pieces and coarsely grate the other half.
Add celery and chopped half of the carrot to pot and cook until beginning to soften.
While broth is heating trim ends and peel any large broccoli stalks and chop into 1/2" dice. Reserve about 1/2 cup of chopped florets. Add the rest of the broccoli florets and stalks to broth.
Cook until all vegetables are softened.
Remove pot from heat and with a stick blender or blender carefully puree soup to both thicken it and to break down any large chunks of vegetables. This does not need to be perfectly smooth though it could be if you like your soups very smooth.
Return soup to pot (if you've used a blender) and to the heat.
Add cream, sour cream, remaining grated carrot and reserved broccoli florets to the pot. Heat at medium for about another 8-10 minutes until second batch of broccoli and carrot are just cooked. Stir regularly to prevent the cream from scorching.
If you do not have smoked cheddar you can just use 2 1/2 C of sharp cheddar.
*This recipe should serve 2 large bowls for a light dinner or 4 smaller bowls for lunch or a starter.