Boil the Ham Bone (optional). Place your ham bone in a large soup pot and cover it with enough water to fully submerge the ham bone (about 12 cups). Cover the pot and bring water to a boil over high heat. Once boiling, reduce heat to low and simmer for at least 1 hour (see notes).
Cook the vegetables. Heat a large pot or Dutch oven over medium heat. Add the olive oil and the onions and mix to coat. Cook for 5-6 minutes or until the onions start to soften. Add the chopped carrots and celery and cook for 10-15 minutes, stirring occasionally. In the last 30 seconds stir in the minced garlic and mix well to combine.
Add the seasonings and split peas. Once the broth is ready, add the bay leaves, green split peas, thyme, and black pepper.
Strain and add the homemade ham broth. Use kitchen tongs to carefully remove the ham bone to a clean plate to cool. Strain approximately half of the broth (about 6 cups) through a fine-mesh strainer into the pot with the split peas, then strain the remaining broth into a separate pot or bowl and save it for later. Add an additional 4-6 cups of water or low-sodium beef broth to the pot, mixing well to combine. See notes.
Remove the meat and add it to the soup. Remove as much meat from the pork bone as possible. Add the meat and bone back to the pot.
Simmer. Bring to a boil, then reduce the heat to low. Cover and simmer for at least 2 hours, stirring occasionally as the peas soften, thicken, and settle at the bottom of the pot.
Serve. When ready to serve, season with additional salt and pepper to taste and discard the bay leaves. Serve with croutons, Tabascos sauce, fresh parsley, or cheese, if desired. Enjoy!