Here are a couple recipes for this standard.
My parents always made split pea soup with sausage instead of ham (polish sausage usually), cut up into little rounds. I've discovered this appears to be a version associated with the Dutch, and in fact my mother grew up in an area with a very strong Dutch ancestry (there's even a nearby town called "Holland" that has an imported windmill). I still make it like my parents did (it's very good) but here are more standard recipes.
Regarding consistency - the Cook's Illustrated recipe says to simmer until the peas pretty much fall apart, whereas the Washington Post one stops before they reach this point but says if you like the soup smooth then to puree it with an immersion blender (before adding in the ham). WE always just kept simmering it until the split peas dissolve altogether. It ends up with a thick creamy texture, in fact leftovers will have the texture of porridge and I usually thin it out at that point.
Washington Post
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Cooks Illustrated
Here are the recipe notes:
Instead of the traditional hambone, we opted for a ham steak. Simmering the steak and some bacon in the broth made our Split-Pea and Ham Soup recipe smoky and meaty. Unsoaked peas absorbed the pork-enriched broth so that every layer of the soup was deeply flavored. Gently fried croutons floated on top of the split-pea soup were the perfect finish to this updated classic.
Four ounces of regular sliced bacon can be used, but the thinner slices are a little harder to remove from the soup. Depending on the age and brand of split peas, the consistency of the soup may vary slightly. If the soup is too thin at the end of step 3, increase the heat and simmer, uncovered, until the desired consistency is reached. If it is too thick, thin it with a little water. Serve the soup sprinkled with Buttery Croutons (related), fresh peas, and chopped mint and drizzled with aged balsamic vinegar. The soup can be made up to 3 days in advance. If necessary, thin it with water when reheating.
Ingredients
2 tablespoons unsalted butter
1 large onion, chopped fine (about 1 1/2 cups)
Table salt
2 medium garlic cloves, minced or pressed through garlic press (about 2 teaspoons)
7 cups water
1 ham steak (about 1 pound), skin removed, cut into quarters
3 slices thick-cut bacon (about 4 ounces) (see note)
1 pound green split peas (about 2 cups), picked through and rinsed
2 sprigs fresh thyme
2 bay leaves
2 medium carrots, peeled and cut into 1/2-inch pieces (about 1 cup)
1 medium celery rib, cut into 1/2-inch pieces (about 1 cup)
Ground black pepper
Instructions
1. Heat butter in large Dutch oven over medium-high heat. When foaming subsides, add onion and ½ teaspoon salt; cook, stirring frequently, until softened, about 3 to 4 minutes. Add garlic and cook until fragrant, about 30 seconds. Add water, ham steak, bacon, peas, thyme, and bay leaves. Increase heat to high and bring to simmer, stirring frequently to keep peas from sticking to bottom. Reduce heat to low, cover, and simmer until peas are tender but not falling apart, about 45 minutes.
2. Remove ham steak, cover with foil or plastic wrap to prevent drying out, and set aside. Stir in carrots and celery; continue to simmer, covered, until vegetables are tender and peas have almost completely broken down, about 30 minutes longer.
3. When cool enough to handle, shred ham into small bite-size pieces with two forks. Remove and discard thyme sprigs, bay leaves, and bacon slices. Stir ham back into soup and return to simmer. Season to taste with salt and pepper; serve.