Optionally peel the potatoes and carrots. If you don't peel them, give them a good scrub. Chop them into a 3/4" dice.
Chop the ham into a smaller dice, about 1/2"
In a soup pot, saute the onions in the olive oil with a teaspoon of salt. Use enough olive oil to make the onions glisten.
When the onions are soft and translucent, add the garlic and oregano and several cranks of black pepper from the pepper grinder. Saute another minute or two.
Add the diced root vegetables along with half of the split peas and the stock. Turn up the heat and bring to a boil, then turn down the heat to a simmer. Continue to simmer, uncovered.
Stir occasionally, using a wooden spatula to scrape the bottom of the pot to loosen anything that tries to stick.
After 40 minutes, taste for seasoning. You used some salt with the onions, and the ham and stock can provide varying levels of salt. The broth should taste well seasoned at this point. Add salt as necessary. When salted to taste, add the remaining half pound of split peas.
Continue to stir occasionally. The scraping of the bottom of the pan becomes more important as the soup thickens. If it seems that things have gotten away from you, don't panic. Remove the pot from heat and let it sit for a minute or two. Often the liquid above the crusty layer will loosen it up in the absence of heat and allow you to scrape it up and work into the soup.
After another 40 minutes, the first batch of split peas should have disintegrated into a thick broth and the second batch have cooked to tender-firm. Taste for texture and seasoning. Make final adjustments to seasoning, and serve when the texture is to your liking.