Add olive oil to a big pot over med-high heat. Stir in onions and salt and cook until the onions soften, just a minute or two. Add the split peas and water. Bring to a boil, dial down the heat, and simmer for 20 minutes, or until the peas are cooked through. Using a large cup or mug ladle half of the soup into a bowl and set aside. Using a hand blender (or regular blender) puree the soup that is still remaining in the pot. Stir the reserved (still chunky) soup back into the puree - you should have a soup that is nicely textured. If you need to thin the soup out with more water (or stock) do so a bit at a time. Stir in the lemon juice and taste. If the soup needs more salt, add more a bit at a time until the flavor of the soup really pops.
Ladle into bowls or cups, and serve each drizzled with olive oil and topped with a good pinch of smoked paprika and a touch of lemon zest.
Serves 4 to 6.