Melt the butter in a large soup pot over medium heat. Add the diced onion and cook until softened. Stir in the garlic and cook briefly until fragrant.
Add the zucchini, rosemary, thyme, celery salt, black pepper, salt, and optional cayenne. Cook for about 5 minutes, stirring occasionally.
Pour in the chicken broth and soy sauce or Worcestershire sauce. Add the chopped potatoes and stir everything together.
Bring the soup to a boil, then reduce the heat to a gentle simmer. Cook for about 20 minutes, or until the potatoes and zucchini are very tender.
Remove the pot from the heat. Carefully blend the soup with an immersion blender until smooth and creamy.
For a richer Creamy Zucchini Soup, reduce the heat to low and stir in the half-and-half or heavy cream.
Add the shredded cheddar gradually and stir until completely melted. Avoid boiling the soup after adding the dairy.
Taste and adjust the salt, pepper, or other seasonings as needed. Serve warm with croutons, crusty bread, fresh herbs, or a sprinkle of cheese.