Split Green Pea, Potato and Onion Soup

6 cups low-sodium canned beef broth, or as needed
1 1/4 cups split green peas, picked over
2 medium potatoes, peeled and cut into 1/2-inch cubes
4 tablespoons (1/2 stick) unsalted butter
2 tablespoons olive oil
1 large onion, cut into 1/2-inch dice
1/4 teaspoon fennel seeds
1 medium carrot, peeled and cut into 1/4-inch dice
1/4 cup freshly grated Parmigiano-Reggiano cheese
Salt and freshly ground black pepper, to taste

1. Put 4 cups of the broth into a large soup pot and add peas and potatoes. Bring to a boil over medium-high heat, then reduce the heat to a very slow but steady simmer. Cook covered, until the peas and potatoes are thoroughly tender, about 45 minutes.
2. Meanwhile, heat the butter and oil in a large skillet over medium heat. Add the onion and sauté until it begins to soften, about 4 minutes. Add the fennel seeds and continue cooking until the onion begins to turn golden, 4 minutes more. Remove from the heat and add to the potatoes and peas.
3. If you prefer a smooth soup- Purée the pea and potato mixture in batches, using a food processor or food mill, and return it to the pot along with 1 cup more of the broth. Otherwise just leave it for a chunkier soup. Add the carrot and cook over low heat, covered, until the carrot is just tender, 8 to 10 minutes. If the soup seems too thick, add some or all of the remaining broth.
4. Remove the soup from the heat. Stir in the grated cheese, and season with salt and pepper. Serve in warmed bowls, adding a teaspoon of the Onion Garnish to each serving. Serve the remaining garnish alongside.
5. If you have ham pieces add those and in the soup until warmed.