Separate broccoli tops from stems. Divide the top broccoli flowers into bite sided pieces and set aside for later in this recipe (roughly 3 cups). Discard the tough, pithy bottom stem. Coarsely chop the remaining stems, should yield roughly 1-2 cups
In a 3-quart pot or dutch oven, over medium heat, saute broccoli stems, onion, garlic and celery in roughly 2 tablespoons of butter until soft.
Add bouillon, mustard powder, white pepper, sea salt and water, mix well to combine. Simmer over medium heat until the vegetables are soft.
Carefully transfer cooked mixture to your blender along with 2 tablespoons starch and puree until completely smooth.
Set pureed mixture aside for a few minutes while sautéing broccoli flowers and carrots with 2 tablespoons of butter over medium heat in your (dirty) pot for about 2-3 minutes.
Add pureed mixture to the broccoli and carrots, reduce heat to low, cover and cook for 5 minutes to lightly cook broccoli and carrots.
After 5 minutes, remove lid and add in the half & half. Keep heat on low and cook for an additional 5 minutes to fully cook carrots. Season with additional white pepper and sea salt to taste. For a thicker soup, puree about 1 cup of the liquid with an additional 1-2 tablespoon starch.
Serve bowls of creamy broccoli soup topped with shredded cheddar cheese, if desired.