Heat ¼ cup extra-virgin olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion, chopped carrot, and chopped celery and cook, stirring occasionally, until softened, about 8 minutes.
Add the chopped garlic, 1 tablespoon chopped fresh rosemary, 1 teaspoon sea salt, ½ teaspoon dried thyme, and several grinds of freshly ground black pepper. Cook, stirring, until the garlic and herbs are fragrant, about 1–2 minutes.
Add the 4½ cups cooked white beans (drained and rinsed) and 4 cups vegetable broth. Bring the mixture to a simmer and cook, uncovered, for 20 minutes.
Working carefully with hot liquid, transfer half of the soup to a blender and blend until smooth; alternatively, use an immersion blender to purée about half the soup in the pot. If using a blender, blend in batches and vent the lid slightly to avoid pressure buildup. Return the purée to the pot and stir to combine.
Add the 6 cups chopped kale leaves and 2 tablespoons fresh lemon juice. Cook, stirring, until the kale is wilted and tender, about 5 minutes.
If the soup is too thick, stir in additional vegetable broth, up to ½ cup at a time, until it reaches your desired consistency.
Taste and adjust seasoning with more sea salt and freshly ground black pepper, if desired.
Serve the soup with a drizzle of extra-virgin olive oil, homemade croutons, and fresh parsley as garnish.