Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
Roast the veggies: Place the cauliflower florets on the baking sheet, drizzle with 1-2 tablespoons of olive oil, and season generously with salt and pepper. Toss to coat.
For the garlic, peel and discard the outer papery layers, leaving the skins of the individual cloves intact. Slice off ¼ inch from the top of the garlic head to expose the cloves. Drizzle with olive oil and loosely wrap in foil. Place the garlic on the sheet with the cauliflower.
Roast the cauliflower and garlic for 30-35 minutes, flipping halfway through, until the cauliflower is golden and caramelized.
Sauté the onion: While the cauliflower is roasting, heat 1 tablespoon of olive oil in a pot over medium heat. Add the diced onion and sauté for 5-8 minutes until softened and translucent.
Blend the soup: Once the cauliflower and garlic are done roasting, transfer them to a blender along with the sautéed onion, broth, salt, and pepper. Allow the garlic to cool, then squeeze the roasted garlic cloves out of the skins into the blender. Blend until smooth (about 1 minute).
Simmer and finish: Pour the blended soup back into the pot and place over medium heat. Bring it to a light simmer, then stir in the shredded cheddar cheese. Let it simmer for 10-15 minutes, stirring occasionally. Taste and adjust the seasoning as needed.
Serve and garnish: Ladle the soup into bowls, and garnish with sliced green onion, extra cheddar cheese, and croutons or toasted sourdough bread. Enjoy!