Trim the tough green tops of the leeks. Cut off the root and slice in half crosswise. Cut into very thin long strips. Place in a bowl with cold water and swish them around to loosen any dirt
Put the olive oil and shallots in a 12-inch skillet and place over medium heat Saute until the shallots become lightly colored, about 5 minutes. Add the red onions and leeks, season generously with salt and pepper. Add about ¼cup of water, turn the heat down to medium low and cover the pan. Cook until the onions and leeks have wilted completely and are very tender, 20 to 30 minutes.
Fill a pot for the pasta with about 6quarts of water, cover, and place over high heat. When the water boils, add 2tablespoons salt, add the spaghettini, and stir until all the strands are submerged. Cook until al dente.
In the meantime, uncover the skillet and raise the heat to medium high. Cook, stirring occasionally, until any liquid in the pan has evaporated and the onions and leeks begin to color. Add the wine and parsley and cook until the wine has evaporated almost completely.
When the pasta is done, drain well, toss with the sauce and the Parmigiano-Reggiano and serve at once.