Preheat your oven to 400°F (200°C).
Arrange the cauliflower florets, diced potatoes, and quartered onion on a baking sheet. Place the garlic head (cut side up) among the vegetables.
Drizzle the vegetables with olive oil, season with salt, and toss to coat evenly.
Roast in the preheated oven for 25–30 minutes, flipping halfway, until the vegetables are tender and golden brown.
Transfer the roasted vegetables to a large pot. Squeeze the roasted garlic cloves out of the skin into the pot and discard the skins.
Add the broth and half-and-half to the pot. Use an immersion blender to puree the soup until smooth and creamy. Alternatively, blend the soup in batches using a blender.
Heat the soup on medium-low until it gently boils, then reduce to a simmer. Cover and let simmer for 5–8 minutes to meld the flavors.
Stir in the shredded cheddar until melted and smooth. Add the thyme, salt, and pepper to taste. Adjust the consistency with extra broth if needed.
Garnish with additional thyme or cheddar, if desired, and serve warm.