Clean and dice all the vegetables as I explained above;
Finely chop the onion, garlic, parsley, and celery;
Heat a little extra virgin olive oil in a big stockpot over high heat and brown the chopped vegetables;
Add all the cleaned vegetables to 2 ½ liters of water and bring it to a boil;
Cook it over high heat for a few minutes;
Then reduce the heat to low; cover the pot and simmer;
Stir the soup frequently to prevent the vegetables from sticking to the bottom of the pot;
Add a little water if necessary;
Halfway through cooking, add the Parmigiano rind cut into small pieces, and a handful of coarse salt;
Coarsely mash the potatoes and the beans with a spoon to gradually thicken the soup; you may also use a fork. Mash some but not all of the potatoes and beans against the side of the pot;
When all the veggies are tender and the soup is thick and creamy, add the pasta;
If it gets too thick, add some water;
Cook the pasta al dente according to the instructions on the package, remove from the heat, and add the pesto. Stir it with a wooden spoon until evenly combined;
Pour your soup into single serving bowls, drizzle each with a little extra virgin olive oil. Sprinkle each bowl with Pecorino or Parmigiano cheese to your taste. Top each with a few fresh basil leaves.