In a skillet over medium heat, saute the onions in half of the olive oil until caramelized, about 30 minutes. Remove half of the onions from the skillet and set aside. Add the remaining olive oil and continue cooking over high heat until the onions are crispy and golden brown; add more olive oil if necessary. Drain on paper towels, season with salt and pepper and set aside for garnish.
In a small pot of salted simmering water, poach the chicken livers for 5 minutes on low heat. Drain on paper towels.
In a food processor, puree the livers with the carrots, 2 hard-boiled eggs and the caramelized onions until smooth. Season with salt and pepper. Add nutmeg. Push through a fine mesh sieve if desired.
Pack the mousse into a serving dish and smooth the top. Cover with plastic wrap. Refrigerate for at least 2 hours.
Garnish with the crispy onions and grate the remaining 2 hard-boiled eggs right before serving. Serve with bread, toasts or crackers.