Rinse the split peas under cool running water and pick out any debris.
In a large pot, sauté onion, garlic, carrots, and celery until softened.
Add split peas, ham bone or diced ham, and stir to coat the peas with flavor.
Pour in the chicken broth or water, add bay leaf and optional spices.
Bring to a boil, then lower heat and simmer uncovered for 60–90 minutes, stirring occasionally.
Remove the ham bone, shred remaining meat, and return it to the pot.
Adjust the thickness by adding water or broth if needed.
Season with salt and pepper, remove bay leaf, and serve warm.