In a large mixing bowl, combine ground beef, grated onion, rinsed arborio rice, parsley, dill, one egg, oregano, salt, and pepper. Mix gently with your hands until just combined, being careful not to overwork the meat. Shape the mixture into golf ball-sized portions (about 16 to 18 meatballs) and place them on a plate. Chill in the refrigerator for 15 to 20 minutes.

In a wide-bottomed pot, bring chicken broth and olive oil to a gentle simmer over medium heat. Carefully add the chilled meatballs to the simmering broth, making sure they have enough room to cook evenly. Cover the pot and reduce heat to low, allowing the meatballs to cook for 30 minutes until tender throughout and the rice inside is fully cooked.

While the meatballs simmer, prepare the avgolemono sauce by whisking together two eggs and lemon juice in a medium bowl until well blended. Slowly ladle about one cup (roughly 3 to 4 ladles) of hot broth from the pot into the egg mixture while whisking constantly. This tempering process prevents the eggs from scrambling.

Remove the pot from direct heat and slowly pour the tempered egg-lemon mixture back into the pot while stirring gently. Return to very low heat and stir continuously for 2 to 3 minutes until the sauce thickens slightly and coats the meatballs. Never let it boil after adding the sauce to prevent curdling.

Serve warm with additional fresh dill or lemon wedges if desired. Pair with rice, orzo pasta, or crusty bread to soak up the delicious sauce.