1. Preheat the oven to 425°F. Chop the broccoli into florets and arrange on a sheet pan (or two) with the garlic cloves. Drizzle 2 tablespoons of the olive oil over the broccoli and garlic. Sprinkle the broccoli with sea salt and pepper. Place in oven and roast for 10 minutes, stirring the broccoli around half way through.
2. Heat remaining 1 tablespoon of oil in a large pot over medium heat and add the chopped onion. Cook for several minutes, until soft. Add the potato along with the water or stock to the pot and bring the mixture to a boil.
3. After 10 minutes of roasting, remove broccoli and garlic from the oven and add to the soup pot. Bring to boil again. Reduce heat to a simmer, and cook for 5-10 more minutes, until the vegetables are very tender when pierced with a fork. Remove from the heat and allow to cool a bit.
4. Transfer soup to a blender (or purée right in the pot if you have an immersion blender) and purée in batches until smooth. Return the soup to the pot and add the cheese. Stir until it has melted completely. Add crème fraîche and stir to combine again. Taste and adjust seasonings, and add more water or stock if you want to thin the soup a bit. Or, if you'd like a thicker, cheesier soup, add more cheese. Serve individual bowls with additional crème fraîche swirled in, if you like; freshly ground pepper is also a "must" on top.