Wash and chop the celery, onion, potato, and garlic.
Heat olive oil or butter in a large pot over medium heat. Add the onion and cook for 4-5 minutes until softened.
Add the garlic and cook for 1 minute until fragrant.
Add the chopped celery and diced potato. Season with salt, black pepper, and dried thyme. Cook for 3-4 minutes.
Pour in the vegetable broth and bring to a gentle boil.
Reduce the heat, cover, and simmer for 20-25 minutes, until the celery and potato are completely tender.
Remove from the heat and blend with an immersion blender until smooth and creamy.
Stir in the heavy cream or milk if using.
Add the fresh lemon juice and adjust the seasoning to taste.
Serve warm and garnish with fresh parsley or celery leaves.