Mix the meatball ingredients together in a bowl until combined. Shape the meatballs into walnut sized balls. Place the meatballs on a clean plate, cover with plastic and refrigerate until you are ready to cook them.
Place the dry chilies, garlic cloves and tomato into a small sauce pan. Fill with water until the chilies are completely covered. Boil the chilies and tomato on medium high heat for 15 minutes. Cover and set aside.
In a large stock pot, heat the oil. Saute the onions for 2 minutes. Add in the carrots and stir. Pour in the water, cumin, bouillon cubes, chayote, potatoes and green beans.
While the soup is heating up, place the tomato, chilies and garlic in the blender with 2 cups of fresh water. Blend on high or 5 minutes.
Using a fine mesh strainer, strain the blended chile into the stock pot scraping the bottom with a spoon. This may require two or three times (seeds and skins will get stuck so you may have to do this in two or so batches rinsing the sieve clean after each time.) Once you have strained all of the chile into the stock pot, stir and bring to a boil. Once the soup is boiling taste the soup and add salt if needed. Cook the soup for 15 minutes.
Next, turn the heat down to medium low. Add in the meatballs one at a time until they are all in the pot. Gently stir the soup to make sure the meatballs are incorporated in with the broth and vegetables. Cover the soup and cook for an additional 30 minutes.
Serve the soup in bowls garnished with chopped cilantro and lime wedges.