Slice the cured pork into small, matchstick thick, pieces
Heat a soup pot over medium heat. Add the pork and stir in some olive oil. A tablespoon is probably enough. You want to make the pork and the pot bottom glisten, but you don't want a pool of oil. Sauté until the pork has visibly rendered it's fat. Expect about 5 minutes.
Add the onion, carrot and celery to the pot along with some salt (about 1 tsp). Stir. If the aromatics glisten, then there is enough fat. If they appear dry, add another splash of olive oil. Increase the heat to medium-high and sauté, stirring frequently, until the carrots are soft and the onions are translucent. Expect 4-5 minutes.
Add the garlic and stir into the aromatic mixture. Make an open well in the center of the pan and add the tomato paste directly on the metal of the pan. Let the tomato paste sizzle on it's own for a minute and then stir everything back together.
If a fond (browned bits stuck to the bottom of the pot) has developed, start with a good splash of the broth to deglaze the pot, scraping with a wooden spatula.
Add the remaining stock and the greens and the beans to the pot. Turn the heat on high until it boils. Turn down to a simmer and let simmer for 15-20 minutes.