Add peeled and chopped potatoes to a large pot, then pour in the vegetable broth. Bring to a boil over medium-high heat.
Reduce the heat to medium and let the potatoes simmer until they are tender and easily pierced with a fork, about 15 minutes. The broth will become slightly cloudy as the potatoes cook.
While the potatoes cook, heat a small pan and gently sauté the minced garlic until fragrant, about 1 minute, until it releases a warm, savory aroma.
Add the sautéed garlic and smoked paprika to the soup pot, stirring well to combine and infuse the broth with flavor.
Use an immersion blender directly in the pot or transfer the soup in batches to a blender, then blend until smooth and creamy. Be careful with hot liquids!
Stir in the coconut milk for added creaminess, and season with salt and pepper to taste. Warm the soup gently over low heat, just until heated through.
Check the seasoning and adjust with more salt, pepper, or spices if desired. The soup should be velvety, with a rich, inviting aroma.
Serve hot, garnished with fresh herbs or a drizzle of coconut milk if you like. Enjoy this cozy, nourishing vegan potato soup!