Place split peas, ham bone (with leftover meat on it), onion, celery, carrots, garlic, thyme, basil, bay leaves, cayenne and water or broth in a large heavy pot.
Bring to a boil skimming off any foam with a spoon.
Reduce heat to simmer and place a lid on your pot.
Make sure the lid is slightly ajar to allow a little steam to escape. Stir occasionally.
When peas are soft and soup is thick, about 2-1/2 hours, remove from heat.
Remove ham bone from the soup; strip any meat from bone, chop it and return to the soup. Discard bone.