To make the meatballs, heat the oil in a small skillet over medium-high heat.
Add the onion and saute until soft, about 5 minutes.
Add the garlic and saute 1 minute longer.
Transfer to a bowl and stir in the remaining ingredients.
Working with about 1 tsp. of the mixture at a time, roll it between the palms of your hands to form balls.
Place on a plate, cover loosely, and refrigerate for about 20 minutes before cooking.
In a saucepan, bring 4 cups of the stock to a boil over medium-high heat.
Add as many meatballs at a time as will fit comfortably in the pan and cook, skimming off foam as necessary, until done, about 5 minutes.
Remove the meatballs with a slotted spoon and set aside; cook the remaining meatballs in the same way.
Strain and reserve the cooking liquid for another use.
In a soup pot or large saucepan, combine the remaining 2 quarts of stock, carrot, chayote, potatoes, and mint.
Bring to a boil over medium-high heat, then reduce to low and simmer, uncovered, until the vegetables are tender, about 25 minutes.
Meanwhile, cook the pasta in a large pot of boiling water until al dente.
Drain and set aside. About 5 minutes before the vegetables are done, heat through the meatballs and pasta.
Season to taste with salt and pepper.
Ladle into preheated bowls and garnish with herb sprigs.
Alternatively, pour into a container and refrigerate, uncovered, until cool, then tightly cover and store up to 3 days.
Slowly reheat before garnishing and serving.
Serves 6 to 8 as a soup course OR 3 to 4 as a main course.