Melt butter in a large soup pot over medium heat. Add diced onion and cook until soft, about 5 minutes.
Stir in minced garlic and cook for 1 minute until fragrant.
Add chopped zucchini, dried rosemary, thyme, celery salt, black pepper, salt, and optional cayenne pepper. Cook for 5 minutes, stirring occasionally.
Add diced potatoes, chicken broth, and soy sauce or Worcestershire sauce. Bring to a boil, then reduce heat to simmer partially covered for about 20 minutes until vegetables are tender.
Remove soup from heat and blend until smooth using an immersion blender or in batches using a regular blender.
Return soup to low heat. Stir in half-and-half or heavy cream.
Gradually add shredded cheddar cheese, stirring until fully melted and smooth.
Serve soup warm for a creamy and comforting experience.