Heat the olive oil in a pot over medium heat. Add the diced pancetta and cook it until most of its fat has rendered. The bottom of the pan should be coated.

Using a slotted spoon, remove the pancetta (leaving the fat in the pan) and transfer it to a food processor. Give it a few good pulses until it looks like a paste. Set it aside.

In the same pot that you cooked the pancetta, add the onions, celery, carrots, and sprigs of thyme. Cook it all together over medium-high heat for about five minutes. Add the pancetta back to the pot, give it all a good stir, and let it cook for another few minutes.

Next, add the potatoes, cauliflower, vegetable broth, tomatoes, and 2 cups of water. Add 1 teaspoon of salt and stir everything together. Bring it to a boil, reduce the heat, and simmer the soup for about 45 minutes.

Remove the thyme stems and discard them. Add the beans, chopped spinach, and parsley. Simmer the soup for 15 minutes. Check the soup for seasoning and add additional salt and pepper to taste.

Ladle the soup into bowls and top each with grated pecorino romano cheese if using.