In a skillet, caramelize the shallots in the butter. Season with salt and pepper. Deglaze with the wine and reduce until almost dry. Let cool.
In a bowl, combine the shallots, chives and duck. Set aside.
In another bowl, combine the eggs and milk. Season with salt and pepper. In two piping hot 30-cm (12-inch) non-stick skillets, melt the butter. Divide the egg mixture between the skillets. Cook over medium heat for 3 to 4 minutes, breaking the bottom of the omelets with a spatula to facilitate cooking. When the edges of the omelet, and not the centre, are almost cooked, spread the filling and cheese over one half of the omelets. With a spatula, fold the other side over the filling. Cut in half and slide into plates. Serve with a green salad.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.