Ham and White Bean Soup

Ingredients

2 yellow onions, divided
4 large celery stalks, divided
4 carrots (about 1 pound), peeled and divided
10 garlic cloves, peeled and divided
2 tablespoons extra-virgin olive oil, divided
1 (2- to 3-pound) smoked ham bone, preferably with meat
8 ounces Parmigiano-Reggiano cheese rinds or pecorino Romano cheese rinds (4 to 5 rinds)
7 thyme sprigs
12 cups water
1 tablespoon coarsely ground black pepper, plus more for serving
1 pound (2 1/2 cups) dried white beans (such as Great Northern beans or cannellini beans), soaked overnight, drained
1 ounce pecorino Romano cheese or Parmesan cheese, finely shredded (about 1/4 cup), plus more for serving
Kosher salt, to taste