Heat oil in a large pot over medium heat; cook onion, covered, until soft, 4 to 5 minutes. Add garlic and a pinch of salt, and cook for 3 minutes more. Add cauliflower, and pour in filtered water until it reaches just below the top of the cauliflower.
Bring to a boil over high heat. Add 2 tablespoons dill. Reduce heat to low, and simmer until cauliflower is just tender, about 10 minutes. Stir in greens, and simmer for 3 minutes.
Let sit for 5 minutes to cool slightly. Stir in remaining 2 tablespoons dill. Puree soup in batches in a blender until very smooth, adding more water (about 1/2 cup) if it's too thick. Return to pot, and reheat. Season with salt and pepper. Garnish with dill, black pepper, a drizzle of oil, and pinch of sea salt.
This soup is fantastic and so incredibly creamy it's hard to believe there isn't any milk or cream. I did add more greens than what is called for (beet greens and chard) and used vegetable broth instead of plain water (a bit extra too, to accommodate the extra veggies). Excellent soup, definitely one we'll make again and again!