Set your oven to 400°F (200°C). Line a baking sheet with parchment paper.
Arrange cauliflower florets on the baking sheet. Drizzle with 2 tbsp olive oil, sprinkle with salt and pepper, and toss to coat. Trim the top of the garlic head, drizzle with 1 tbsp olive oil, wrap it in foil, and place it on the baking sheet with the cauliflower. Roast for 30-35 minutes, until cauliflower is golden and garlic is softened.
While roasting, heat the remaining olive oil in a large pot over medium heat. Sauté the onion until translucent, about 5-8 minutes.
Add the roasted cauliflower and squeezed roasted garlic cloves to the pot with the onions. Pour in the vegetable broth, then use an immersion blender to puree until smooth. For a countertop blender, work in batches, blending until creamy.
Return the blended soup to low heat. Gradually stir in the shredded cheddar until melted and fully incorporated. Taste and adjust seasoning with salt and pepper.
Ladle the soup into bowls and garnish with green onions, extra cheddar, or croutons as desired.
