Add only the 4 cups chopped potatoes to a medium pot. Add the water, salt, pepper, onion, garlic, and smoked paprika. Stir well and bring to a boil. Once boiling, cover and turn to low and simmer for 15-20 minutes until the potatoes are really tender.

Preheat the oven to 375?F and line a pan with parchment paper. Spread the broccoli out evenly and lightly season with salt and pepper. If baking extra potatoes for garnish, add them to a separate pan, season with salt and pepper and place them on the bottom rack.

Bake the broccoli on the middle rack for 15 minutes and remove. Leave the potatoes in the oven for 10 minutes or more until golden brown. After you remove the broccoli, move the potatoes to the center rack to continue cooking.

Once the potatoes in the pot are tender, add the coconut milk, nutritional yeast, and pickle juice. Using a handheld immersion stick blender, blend until all the potatoes are smooth and the soup is creamy and thick. Taste and adjust spices if necessary.

Stir in the broccoli and top with extra baked potatoes.