Peel and chop the potatoes into roughly 1-inch cubes, setting them aside.
Dice the onion and mince the garlic, filling your kitchen with a fresh, aromatic scent as you do.
Heat the large pot over medium heat, then add a splash of oil or a little water for a oil-free option. Sauté the diced onion until translucent and soft, about 5 minutes, until it begins to turn golden around the edges.
Add the minced garlic to the onions and cook for another minute, stirring until fragrant—this will fill your kitchen with a warm, inviting smell.
Pour in the vegetable broth and add the chopped potatoes along with a pinch of salt. Bring the mixture to a gentle boil, then reduce the heat to low and cover the pot.
Let everything simmer uncovered for about 20-25 minutes, or until the potatoes are very soft and starting to fall apart, filling the kitchen with a comforting starchy aroma.
Use an immersion blender directly in the pot to puree the soup until velvety smooth, or carefully transfer the hot mixture into a blender in batches, blending until creamy and uniform.
Stir in the plant-based milk and season with salt, pepper, and smoked paprika if using, adjusting to taste. Let the soup simmer for another 5 minutes to meld the flavors together.
Taste the soup and adjust the salt, pepper, or paprika as desired. The soup should be warm, creamy, and fragrant, ready to serve.
Ladle the hot soup into bowls, garnish with fresh herbs if you like, and enjoy this comforting, velvety vegan potato soup.