Heat 1 tablespoon olive oil in a large stalk pot over medium/high heat. Add the onion to the pot and season with ½ teaspoon of salt. Sauté the onion for about 3 minutes.
Add the remaining olive oil, carrots, celery, and mushrooms to the onion with an additional 1 teaspoon of salt. Sauté all of these vegetables for about 4 minutes, and then add the garlic to the pot. Stir and sauté for an additional 2 minutes.
Next, add the split peas, broth, water, mustard, and the remaining salt to the pot and stir until all the ingredients are combined. Tie the thyme and sage together with cooking twine and submerge it in the broth.
Lastly, submerge the hambone into the soup. Bring the mixture to a boil over high heat.
Cover the pot and turn the heat to low. Let the soup simmer for 1 hour, stirring periodically.
Remove the hambone from the pot and add the diced ham to the soup. Let the soup simmer for an additional 15 minutes to heat the ham.
Remove the herb bundle from the soup and then remove the soup from the heat and let it sit for 15 minutes, uncovered to thicken before serving.
Serve alone or with warm bread.