In a large pot or Dutch oven, heat 2 tablespoons of olive oil over medium heat. Incorporate the chopped onion, sea salt, and several grinds of black pepper. Sauté the mixture for 6 to 8 minutes until the onion becomes tender and translucent.
Add the minced garlic to the pot, stirring for an additional 2 minutes. Then, mix in the white wine vinegar, cooking for 30 seconds while stirring. Next, pour in the vegetable broth, along with the chopped potatoes and white beans. Bring the mixture to a boil, then reduce the heat and let it simmer for 30 minutes.
After simmering, allow the soup to cool slightly. Transfer half of the soup to a blender, adding the remaining tablespoon of olive oil, Dijon mustard, fresh lemon juice, and smoked paprika. Blend the mixture until it reaches a smooth consistency before returning it to the pot.
Use a potato masher to gently break down some of the potato chunks and beans in the pot. Taste the soup and adjust the seasoning with additional salt and pepper if necessary. Serve hot, garnished with your choice of toppings.