Heat the oil in a large heavy-bottomed pot (5-quart) over medium heat. Add the onion and cook until tender, about 5 minutes. Add the garlic and cook a minute more.
Add the stock, water, and tomato sauce. Bring to a boil and reduce the heat to a simmer. Add the green beans and carrots.
Mix the uncooked rice into the meat on a large bowl, along with the spearmint leaves and parsley, salt, and pepper. Mix in the raw egg. Form mixture into 1-inch meatballs.
Gently add the meatballs to the simmering soup, one at a time. Cover and let simmer for ½ hour.
Add the peas towards the end of cooking. Add a few pinches of oregano and sprinkle with salt and pepper, and a dash of cayenne, to taste.
Ladle into bowls and garnish with chopped fresh cilantro.
