Coarsely chop the remaining cauliflower.
Heat the oil in the Instant Pot. Saute the onions, celery, garlic, coriander, mustard, thyme, and cayenne. Cook, stirring, until vegetables are soft, 4 to 5 minutes.
Alternatively, heat the oil in a large pot or stovetop pressure cooker over medium flame. Saute the onions, celery, garlic, coriander, mustard, thyme, and cayenne. Cook, stirring, until vegetables are soft, 4 to 5 minutes.
Add the vegetable broth, the coarsely chopped raw cauliflower, and the potato. Press the Manual button on the Instant Pot and set the timer for 6 minutes. If using a stovetop pressure cooker, after the pot is up to pressure, set the timer for 6 minutes. When time is up, turn the dial to vent on the Instant Pot, or quick release a stovetop pressure cooker under cold water. Stir in the white beans.
For regular stove-top, bring the broth, coarsely chopped raw cauliflower and potato to a simmer and cook 15 to 18 minutes. When the potato is tender, stir in the white beans.
In a blender, puree 1 1/2 cups of the vegetables and beans along with 1/2 cup of the broth, being careful that the hot soup doesn’t explode out of the blender. (Cover the lid with a towel and hold the lid on tight.)
Pour the blended mixture back into the pot. Stir in the chopped charred cauliflower. Simmer about 5 minutes. Season to taste with salt and and plenty of freshly ground pepper.
To serve, divide the spinach between 4 bowls and ladle hot soup on top. Top with the chopped parsley and the grated cauliflower that has been “pickling” in lemon and salt.