Peel the potatoes, cut into small pieces, rinse in cold water, drain.
Rinse the split peas in cold water and drain.
Put potatoes and peas in a soup pot with 3 cups (720 ml) broth. Cover, bring to a gentle boil over medium heat, and cook until both are tender. Turn off the heat.
Mill the potatoes and peas with their liquid through a food mill back into the pot.
In a small skillet over medium-high heat, cook the onion in the butter and oil, stirring, until the onion turns a rich gold.
Pour the skillet contents into the pot, add the remaining 2 cups (480 ml) broth, cover, and bring to a slow steady boil over medium heat. Cook, stirring now and then, until the fat floating on top is evenly distributed through the soup.
Stir in the Parmesan before turning off the heat, then taste and adjust salt. Serve with crostini and additional grated Parmesan at the table.