Add the split peas to a bowl and cover with water, make sure they are well covered. Leave overnight, or 8 hours to soak at room temperature.
To make with ham (if not using ham, skip to step 5)
Add the ham hock/shank/bone or smoked bacon, carrot, onion, bay leaves, to a pan and cover with 3 liters (12 cups) water. Bring to a simmer and simmer for 1 hour.
After 1 hour, remove the ham shank/hock, vegetables and bay leaves.
To a pan, add the split peas and pour over enough of the ham broth (or water if not using ham) to cover by about 1-inch (2.5cm), about 4 cups (1 liter). Bring to a boil and simmer for 1 hour.
After 1 hour, check to make sure the peas are soft. If not, continue to cook until they are, you may need to add more liquid as needed.
Once soft (you want almost no liquid left). If you have too much liquid left, continue to cook, then stir in the butter and salt (this will depend on how salty your ham is so start off with only a little) . Mash or use a blender to achieve your desired consistency. Note: It will thicken as it cools. Taste for seasoning.