Trim and clean the mushrooms and then slice them into bite-size pieces. Put a few tablespoons of oil in a pan, turn on the heat, and add the mushrooms once the oil is hot. Sprinkle a pinch of salt over them as they cook, but leave them undisturbed until they release excess water and begin to brown. Gently turn them over to sear the other side.
When they are done, remove them to a plate off the stove. Add the onion and garlic and an additional splash of oil if needed. Sauté the onion mixture on medium heat for a few minutes or until softened. Turn off the heat and deglaze the pan with the balsamic vinegar, scraping off any browned bits on the bottom of the pan. Let the pan cool to room temperature.
In a separate pan, dry roast the pine nuts by putting them in a pan over medium heat and toasting them for a few minutes. Keep a close eye on them since they can go from browned to burnt quickly. Set aside to cool.
Put the cooled vinegar-onion mixture in a bowl and add the olive oil. Whisk until blended and season to taste.
Arrange the greens on plates and top with mushrooms, pine nuts and a light sprinkling of Parmesan and dill. Dress gently with the vinaigrette and serve.