In a small skillet, heat the olive oil and butter and saute the mushrooms, turning to brown both sides. Remove the mushrooms to a small plate.
In a medium saucepan, brown the ham or pancetta in a drizzle of olive oil. Remove to a small plate.
Add the remaining olive oil and heat. Saute the minced onion, carrots and celery, along with the salt to the pan and cook until just getting soft. Add the minced garlic and stir for 20 - 30 seconds.
Add the broth, water, potato pieces, browned pancetta or ham, and balsamic vinegar to the saucepan. Add in 1/8 teaspoon or so of crushed black pepper. Simmer on medium low for 40 minutes.
Add the drained white cannellini beans, mushrooms, and half of the chopped parsley and basil. Simmer for 15 minutes. Remove from heat and add in remaining basil and parsley. Add more pepper to taste. Ladle in bowls and top with shaved or grated pecorino or parmesan.