Preheat your oven to 400°F (200°C).
Cut the cauliflower into florets and peel and dice the potato into cubes.
Toss the cauliflower florets and potato cubes with olive oil, salt, pepper, and dried thyme. Spread them on a baking sheet in a single layer.
Roast the vegetables in the preheated oven for 25-30 minutes, or until they are tender and lightly browned.
While the vegetables are roasting, finely chop the half onion and mince the garlic.
In a large pot, add a splash of olive oil and sauté the chopped onion and garlic until softened and fragrant, about 3-4 minutes.
Add the roasted cauliflower and potatoes to the pot along with the vegetable broth.
Bring the mixture to a simmer and let cook for an additional 5 minutes to meld the flavors.
Transfer the soup to a blender (in batches if needed) and blend until smooth and creamy. Return to the pot.
Stir in the Greek yogurt until well combined, heating through on low heat. Do not boil to prevent curdling.
Adjust salt and pepper to taste, then serve warm with an optional extra drizzle of Greek yogurt on top.