Step 1

Melt the butter in a large, heavy pot (about 4 quarts) over medium heat. Add the onion, celery, and carrots, and cook until the onions are soft. Add the bay leaves, thyme, and season with salt and pepper. Add the split peas and stir to combine. Add the broth and the ham hock/ham bone or smoked paprika and bring to a boil.

Step 2

Turn heat down to low, cover, and simmer for 1 hour. Check on the soup. The peas should be creamy and soft, but not so thick that it's like a pea loaf. If it's getting too thick, add a little more broth or water. The soup is ready when the peas are soft, about 1 to 1 1/2 hours cooking time.

Step 3

Using tongs, transfer the ham onto a cutting board and let cool slightly. Remove the meat from the bones, shred, and stir back into the soup (discard the bones and any skin). Taste and season with salt and pepper as needed, as well as more smoked paprika, if using, if you prefer a stronger smoky taste. Remove the bay leaves before serving.