Cut the ham steak into small cubes and set aside in a large bowl.
Chop the onion coarsely, and add to the ham.
Chop the garlic, add to the ham.
Chop the carrots and celery, add to the ham.
slice the mushrooms, add to the ham.
Heat olive oil in the bottom of a large, 8 quart stock pot. When the oil is hot, add the ham, onion, garlic, carrots, celery, and mushrooms. Saute' ingredients until vegetables are partly cooked, stirring frequently. This takes about ten or fifteen minutes.
Add 1/2 gallon of water (give or take). Bring the soup to a boil, then reduce heat to low. Add parsley, and add your favorite dehydrated italian herb blend to taste. Cover the stock pot and let simmer, slowly, for a minimum of three hours. This allows the vegetable and ham broth to develop, and will be very fragrant. Stir once in a while, not not all that often.
After about three hours, turn the stove off but leave the stock pot on the burner, for one hour. This further develops the broth.
Bring broth to boil again, and stir in two pounds (one common, normal sized bag) of dried green split peas. Wait for the soup to come to a rolling boil, then reduce temperature to low. Simmer, stirring once in a while, for an hour or so.
The soup is cooked when the peas have become soft and have broken down into the liquid. They should retain their sweet-ish character. Add salt and pepper to taste as you serve. I find that the ham and garlic add enough salt for me, but other folks might want a little more.
I have also prepared this broth a day ahead of time and let it sit in the fridge overnight, which fully develops the stock and blends its flavors. You can cut the stock simmering time if you want, but the longer you take with it, the better the soup tastes.