Note: There is no need to peel the zucchini for this soup unless yours is jumbo sized, in which case it's possible that the skins are tough and slightly bitter.
Shred the cheese from a block for best results. Set aside to allow to come down to room temperature. This will allow it to melt and combine into the soup smoothly.
Melt butter in a soup pot over medium heat. Add onions, cook until softened, about 5 minutes. Add garlic and cook 1 more minute.
Add diced zucchini and seasonings: (1/4 tsp each of: dried rosemary, thyme, celery salt, kosher salt, pepper, and an optional pinch of cayenne.) Sauté for 5 minutes.
Peel and rinse the potatoes. Cut them into small pieces of equal size so that they cook evenly and quickly.
Add diced potatoes, chicken broth, and soy sauce. Bring to a boil, then reduce heat to medium. Skim up any brown foam from the top if needed.
Let simmer, partially covered, until the vegetables are very fork tender. About 20 minutes.
Remove from heat. Use an immersion blender or transfer to a blender in batches to blend until smooth.