Clean the leeks: cut off the green top parts and slice them lengthwise, separate each leaf and rinse thoroughly under cold water. Dry the leeks well and then roughly chop them.
Using a wooden spoon, saute the leeks in a large skillet with about 3 tablespoons olive oil over medium-high heat, for about 10 minutes, stirring often. Remove from the heat once completely softened and the leeks begin to turn a golden blonde.
While the leeks are cooking, start the meatballs:
In a large bowl, mix together the ground meat, shallot (or onion), chopped parsley, 2 tablespoons bread crumbs, grated Parmesan, and egg yolk. Season the mixture lightly with salt and freshly ground pepper.
Pour the bread crumbs in a pasta bowl and roll in the meatballs as you are preparing them. Set the meat aside on a large plate, do not make a pile; use 2 plates, if necessary.
Return the pan with the leeks to a medium-high heat, add the meatballs and saute gently flipping them once or twice.
Once they are browned, lower the heat and add the broth a few tablespoons at a time. Season with a light sprinkle of salt and some ground pepper, lower the heat to medium-low and cook for about 25 minutes, always making sure that the sauce does not dry up.
This dish makes Debi very happy, she says that this sauce is probably the closest thing that I prepare that actually deserves to be called gravy!
If you finish the broth, but you believe your meatballs need to be cooked a bit longer, use some tap water from the sink.