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 more minutes. 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 Tbsp 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 Tbsp 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.