This started with the classic Joy of Cooking, but I think I've got almost more notes written on the sides of it than the length of the recipe itself, so it's rather highly adapted. As you can tell, it's a very adaptable, forgiving, non-exact recipe!
1 c. split peas
4-6 c. chicken broth (or water + chicken bouillon*)
1-2 c. chopped ham and/or a bone and/or whatever leftover ham you've got
2 tsp. Worchestershire sauce
spices as desired - maybe some rosemary, a bay leaf, pepper...
a couple of carrots, sliced into coins
a couple of potatoes, diced into middlish small chunks
1. Rinse the peas and make sure there are no stones or junk. Simmer them in a big pan with the chicken broth for like an hour, or until they're soft. Pour most of the soup in the blender and blend until smoothish. You could do it all, but I like a little texture. If you've got one of those immersion blender thingies, it'd work great. Stir in the Worchestershire (for the umami, you know) plus any spices you'd like. If it seems a little thick, just thin it down a bit with water or more stock.
2. In a separate pan, (at any time during this process), saute up the carrots until tender, set aside, and do the same with the potatoes. If you've got leftover roasted or baked potatoes (or carrots for that matter), just cut them up and use those instead with no need to cook 'em.
3. Once the peas are blended, add in the carrots, potatoes, and ham pieces, and put on low heat at least until heated through, but you can let it simmer for hours. If you've got a ham bone, I'd throw it in now and just let it all simmer for a long time, later taking it out and trying to cut off some of the meat.
I've also done this in the slow cooker, putting the peas and broth in on high for a few hours (4?) or until they were so mushy I didn't even have to blend them. Then I added in the pre-cooked carrots and potatoes and ham and turned it to low and let it simmer until supper.
*I now swear by "Better than Bouillon" after using it at my mom's because it actually tastes like soup not just salt.