Chop the broccoli into small florets and set aside then chop the stalk into small pieces keeping the chopped stalk pieces and the florets separate.
Heat the oil in a dutch oven or large pot over medium heat.
Add the onion an garlic and cook, stirring often, until the onion begins to soften, about 5 minutes.
Add the chopped broccoli stalks, potato, 2 tsp salt, and ¼ tsp black pepper to the pot and cook, stirring occasionally, about 1 minute.
Pour the water into the pot then increase the heat to medium-high and bring to a boil.
When the soup begins to boil reduce the heat to medium-low and simmer the soup, stirring occasionally, until the potatoes and broccoli stalks are tender, about 15 minutes.
Take the soup off the heat then use an immersion blender or a stand blender to puree the soup.
Return the soup to the stove over medium-low heat and stir in the broccoli florets then cook, stirring occasionally, for about 3 minutes.
Slowly stir in the heavy cream and cheddar cheese then continue to simmer the soup over medium-low heat for about 2 more minutes or until heated through and cheese has melted.
Taste the soup and season with additional salt & pepper if needed.
