Roast the Broccoli and Garlic : Preheat oven to 200°C (400°F). Toss broccoli with olive oil, salt, and pepper. Cut the top off the garlic bulb, drizzle with oil, wrap in foil, and place on tray. Roast until broccoli is tender and garlic is soft.
Prepare the Soup Base : Heat butter or oil in a pot over medium heat. Sauté onion until soft and translucent.
Blend and Simmer : Add roasted broccoli and squeezed roasted garlic to the pot. Pour in vegetable broth and simmer briefly. Blend until smooth, then stir in milk and Parmesan.