Add olive oil to a big pot over medium-high heat. Stir in onions and salt and cook until the onions soften (approximately 2 minutes). 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. 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 soup back into the puree, for a more chunky texture (or puree all if you prefer a smooth consistency). Stir in the lemon juice and taste. Add a pinch of salt until the flavor is perfect.

Ladle into bowls or cups, and serve each drizzled with olive oil and topped with a pinch of smoked or regular paprika.