Mageiritsa (Easter Lamb Soup)

Ingredients

• intestines, heart, lungs, liver, and head (optional) of 1 lamb
• salt
• 1 ½ quarts water
• ¼ cup extra-virgin greek olive oil
• 1 pound boneless lamb, shredded or finely chopped (optional)
• 8 scallions, trimmed, whites and tender upper greens finely chopped
• 2 cup shredded romaine lettuce
• 1 cup chopped fresh dill
• freshly ground black pepper
• ½ cup short-grain rice, such as greek nihaki or glazed rice
• 5 large eggs, at room temperature
• strained fresh juice of 2 large lemons, or more to taste