Cut 1 onion, 2 celery stalks, and 2 carrots into 1-inch pieces. Smash 6 garlic cloves. Heat 1 tablespoon oil in a large pot or Dutch oven over medium. Add chopped vegetables and smashed garlic; cook, stirring occasionally, until starting to brown, 6 to 8 minutes. Add ham bone, cheese rinds, thyme sprigs, and water; bring to a boil over high. Reduce heat to low; simmer, partially covered, stirring occasionally (to prevent cheese from sticking to bottom of pot), until broth is cloudy and tastes strongly like Parmesan and ham, about 1 hour and 30 minutes.

While broth simmers, chop remaining 1 onion, 2 celery stalks, and 2 carrots into 1/2-inch pieces. Mince remaining 4 garlic cloves. Set aside.

Pour broth through a fine mesh strainer set over a large measuring cup or bowl, pressing vegetables and cheese rinds to extract as much liquid as possible (about 10 cups); discard solids. Reserve ham bone. Wipe pot clean.

Heat remaining 1 tablespoon oil in pot over medium-high. Add pepper and reserved chopped vegetables; cook, stirring often, until vegetables start to soften, 5 to 6 minutes. Add reserved minced garlic; cook, stirring constantly, until fragrant, about 1 minute.

Add reserved ham bone, drained beans, and strained broth; bring to a boil over high. Reduce heat to medium-low; simmer, partially covered, stirring occasionally, until beans are creamy and tender and ham meat easily pulls away from bone, 2 to 3 hours.

Transfer 1 cup of soup (removing any ham pieces) to a small bowl; mash using a potato masher until smooth. (Alternatively, transfer to a blender, and process until smooth.) Stir mashed soup back into pot.

Remove ham bone from soup, and shred meat (about 1 cup); return meat to soup, and discard bone. Stir in pecorino Romano cheese until cheese is melted. Season to taste with salt. Ladle soup evenly into individual bowls; sprinkle with additional black pepper and cheese.