Combine first 4 ingredients in a medium, heavy saucepan, and bring to a boil. Partially cover, reduce heat, and simmer 40 minutes or until the peas are tender and most of the liquid is absorbed, stirring occasionally (the pea mixture will have the consistency of very thick split pea soup). Cool slightly.
Place the pea mixture in a food processor or blender. Add the lemon juice, 2 tablespoons olive oil, cumin, and salt; process until smooth.
Place the pea puree in a shallow dish. Drizzle 1 tablespoon olive oil over the pea puree, and sprinkle with chopped onion and dill.
Note: Store in an airtight container in the refrigerator for up to 4 days.