Preheat the oven to 400°F. Line a large baking sheet with parchment paper.

Roast the cauliflower: Add the cauliflower florets to the prepared baking sheet. Drizzle with 1-2 tablespoons of olive oil and season generously with salt and pepper. Toss to combine.

Prep the garlic: Peel off and discard the outer papery layers of the whole garlic head, leaving the skins on the individual cloves. Using a sharp knife, cut ¼ inch off the top of the head so that the tops of the individual cloves are exposed. Place the garlic on a piece of foil, drizzle the exposed top with olive oil, then loosely wrap the foil around it. Place the foil packet on the baking sheet with the cauliflower.

Roast: Roast the cauliflower and garlic for 30-35 minutes, flipping the cauliflower halfway through, until the cauliflower is golden brown and caramelized and the garlic is soft and fragrant.

Cook the onion: While the vegetables are roasting, heat the remaining 1 tablespoon of olive oil in a large pot or Dutch oven over medium heat. Add the diced onion and sauté for 5-8 minutes, until the onion is soft and translucent.

Blend the soup: Transfer the sautéed onion to a large blender. Add the roasted cauliflower. For the garlic, let it cool slightly, then squeeze the roasted cloves out of their skins into the blender. Add the vegetable broth, salt, and pepper. Blend until smooth, about 1 minute. (If using an immersion blender, add everything directly to the pot and blend.)

Combine: Pour the blended soup back into the pot (if using a regular blender) and place over medium heat.

Add the cheese: Bring the soup to a light simmer, then stir in the shredded sharp cheddar cheese. Simmer for 10-15 minutes, stirring occasionally, until the cheese is fully melted and the soup is hot.

Taste and serve: Taste and adjust seasonings with additional salt and pepper if needed. Ladle into bowls and garnish with sliced green onion and extra shredded cheddar. Serve with crusty toasted sourdough or croutons for dipping.