Warm a Dutch oven over medium-low heat. Drizzle in the olive oil and allow it to heat gently. Add the minced onion, carrot, celery, garlic, and sea salt to the pot. Sauté the aromatics, stirring regularly, until softened and fragrant, about 4 to 5 minutes.
Pour in the chicken stock and stir well to incorporate the vegetables. Cover with a lid and allow the mixture to simmer for 10 minutes to develop depth of flavor.
Stir in the acini di pepe pasta. Replace the lid and let the soup simmer for another 15 minutes, stirring once midway to ensure the pasta doesn’t cling to the bottom.
While the pasta cooks, whisk the egg in a small bowl. Gradually ladle in some hot broth while whisking continuously—this step tempers the egg, preventing it from scrambling.
Slowly pour the tempered egg back into the soup, stirring immediately to distribute it evenly. Cover and let it cook for an additional 5 minutes, allowing the soup to thicken slightly and the egg to fully integrate.
Once finished, remove the pot from heat promptly to keep the pastina from becoming overly soft. Ladle into bowls and finish with a generous sprinkle of Parmigiano Reggiano and freshly ground black pepper, if desired. Serve warm.