In a large pot or Dutch oven, heat olive oil over medium heat. Add chopped onion with sea salt and black pepper, cooking until softened and translucent, approximately 6–8 minutes.
Stir in minced garlic for 2 minutes, then add white wine vinegar, stirring for 30 seconds. Pour in vegetable broth, followed by cubed potatoes and white beans. Bring the mixture to a boil, reduce heat, and simmer for 30 minutes.
Allow the soup to cool slightly. Transfer half to a blender, adding remaining olive oil, Dijon mustard, lemon juice, and smoked paprika. Blend until smooth, then return to the pot.
Gently mash some potato and bean pieces for texture. Taste and adjust seasoning with additional salt and pepper if needed. Serve hot with optional toppings.