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In a small bowl, combine ⅓ cup of the onion slices with the lime juice and a pinch of salt and let sit, tossing occasionally, while the soup simmers.
Heat a large pot over medium-high for a minute or so to warm it up. Add the oil and heat until it thins out, about 30 seconds. Add remaining onion slices, celery, carrot and a pinch of salt, and cook until the vegetables are very soft and brown at the edges, 7 to 10 minutes.
Add the garlic, tomato paste, coriander and pepper, and cook until the paste darkens and the garlic is fragrant, about 1 minute.
Add broth, white beans and kidney beans, and bring to a simmer. Let simmer until the soup is nice and flavorful, adding more salt if needed, 15 to 20 minutes. If you like a thicker broth, you can smash some of the beans with the back of the spoon to release their starch. Or leave the beans whole for a brothier soup.
Add the green beans to the pot and simmer until tender, 4 to 8 minutes.
Top each bowl with some of the pickled onions and a little of the lime juice if you like, plus a generous shower of Parmesan, more pepper, celery leaves, parsley and a drizzle of olive oil.