Sauté aromatics: Heat oil in a large pot over medium heat. Add onion with a pinch of salt and cook 4–5 minutes until soft.
Stir in garlic for 30 seconds.
Add cauliflower and seasonings: Toss in cauliflower, thyme, parsley, smoked paprika, bay leaf, and a bit more salt and pepper. Stir to coat for 1 minute.
Simmer: Pour in chicken broth and bring to a gentle boil. Reduce to a simmer, cover, and cook 12–15 minutes until cauliflower is very tender.
Blend the base: Remove the bay leaf.
Use an immersion blender to puree until smooth. For a blender, carefully blend in batches and return to the pot.
Adjust thickness: Stir in water as needed to reach a creamy, soup-like consistency.
Add chicken and creaminess: Stir in the cooked chicken and Greek yogurt (and Parmesan, if using). Warm over low heat 3–4 minutes.
Do not boil after adding yogurt.
Taste and finish: Adjust salt and pepper. Add a squeeze of lemon or zest if you like brightness. Garnish and serve hot.