Combine chicken stock, ham hocks, onions, potatoes, carrots, split peas, black and white peppercorns, salt, bay leaves, and herbes de Provence in a heavy stockpot.

Bring the mixture to a boil.

Reduce heat to low, partially cover the pot, and simmer for 1 hour, stirring occasionally.

Add 2 cups of water and continue simmering, partially covered, for 1 more hour.

Remove the ham hocks from the pot using a skimmer or slotted spoon.

Slice the meat from the bone and cut it into small bites; set aside.

Remove solids from the pot with a skimmer or slotted spoon, discard the bay leaves, and puree through a food mill or in a food processor.

Return the pureed solids to the pot and blend.

Stir in the remaining 1 cup of water and Marsala wine.

Season with additional salt and white pepper to taste.

Stir in the ham meat, partially cover, and simmer for 30 minutes.

If the soup is too thick, thin it with a bit more water or stock.

Ladle into heated bowls and serve.