Prepare the soup base: In a medium pot, combine the 4 heaping cups of chopped potatoes, water or broth, salt, pepper, onion, garlic, and smoked paprika. Stir well. Bring to a boil over high heat. Once boiling, cover the pot with a lid, reduce the heat to low, and simmer for 15 to 20 minutes, or until the potatoes are very fork tender.
Roast the garnishes: While the soup simmers, preheat your oven to 375°F (190°C). Line a large baking sheet with parchment paper. Spread the broccoli florets evenly on one side of the pan and the remaining 2 cups of extra chopped potatoes on the other side. Season everything lightly with salt and pepper.
Bake: Place the pan in the oven. Roast for 15 minutes, then remove the roasted broccoli from the pan and set it aside. Continue to roast the potatoes for an additional 10 to 15 minutes, or until they are golden brown and crispy.
Blend the soup: Once the potatoes in the pot are completely tender, remove the pot from the heat. Add the "lite" coconut milk, nutritional yeast, and dill pickle juice.
Creamify: Use a handheld immersion blender to blend the soup directly in the pot until it is completely smooth, creamy, and thick. Taste and adjust the seasoning with more salt, pepper, or smoked paprika, if desired.
Finish and serve: Gently stir the roasted broccoli florets into the smooth soup. Ladle the soup into bowls and top generously with the crispy roasted potato croutons. Serve immediately.