Rinse the split peas in a sieve under cold water and pick out any stones.
Heat a large heavy pot over medium heat with a spoon of water or a drop of oil. Add the onion, carrot and celery and cook, stirring, until soft, about 8 minutes.
Add the split peas, broth, water, thyme and half the diced ham.
Bring to a boil over high heat and skim off any foam. Reduce the heat to low, partly cover and simmer for 75 to 90 minutes, stirring every 15 minutes and scraping the bottom, until the peas have broken down into a thick soup. Add water if it gets too thick.
For a smoother soup, blend part of it with a stick blender.
Stir in the rest of the ham, heat through for 5 minutes and season with pepper. The ham is salty, so taste before adding any salt. The soup thickens as it cools; thin leftovers with water.
Tips
Stir every 15 minutes; split peas catch on the bottom.
Swaps
Vegan: skip the ham, use vegetable broth and a pinch of smoked paprika.
Storing and reheating
Keeps 5 days or 4 months frozen. It thickens as it cools; loosen with water.