Stir cauliflower, chicken stock, water and potatoes together in a large pot; bring to the boil. Reduce heat to medium and stir in garlic, salt, white pepper and thyme. Cover pot and simmer until potatoes and cauliflower are soft, 30 to 40 minutes. Remove from heat and let cool slightly, 5 to 10 minutes.
Puree or blend potato and cauliflower mixture until smooth.
Reheat soup and simmer until flavours have blended, about 30 minutes. Adjust salt, pepper and thyme to taste. Don't be tempted to skip this step - the soup might look watery after blending but cooking it further really allows the flavours to develop and the consistency to thicken.