Sort & Rinse Split Peas. To start, sort through the split peas to pick out any brown split peas or any bits of dirt that may have made it into the package. Rinse the split peas in a fine mesh sieve and set aside. (Note: once you've rinsed the beans, they'll immediately start to stick together in a big clump. This is normal and will sort itself out as soon as you add them to the soup.)
Sauté Veggies. In a large Dutch oven or soup pot, heat olive oil over medium heat. Add onion, celery, and carrots, and cook 8-10 minutes, or until softened and tender. Add garlic and stir one minute more, or until fragrant.
Combine. Add diced ham (or ham bone, if using), and rinsed split peas to the pot. Pour in chicken broth and stir to combine. Tuck bay leaf into the soup.
Simmer. Cover the pot and bring the soup to a simmer over medium heat. Reduce the heat to medium-low and simmer 1 1/2-2 hours, stirring every 15-20 minutes or so, until the split peas are tender and the soup has thickened. If your soup is getting too thick, add more broth as you go.
Finish. If using a ham bone, carefully use tongs to remove the ham from the soup and use two forks or a knife and fork to shred or dice the ham off the bone. Return shredded/diced ham to the soup and discard the bone. Remove the bay leaf from the soup and season soup with salt and black pepper, to taste. Garnish with fresh parsley, fresh thyme, crisped bacon, or more black pepper!
Store leftover soup in an airtight container in the refrigerator 3-4 days, or freeze up to 2 months. Keep in mind that split pea soup will continue to thicken significantly when it cools, so you'll want to add a splash of water or broth to thin leftover soup out when reheating.