In a 10-inch skillet, melt half of the butter over medium heat. Raise the heat to medium high. Add the sliced mushrooms and sauté, stirring frequently, until they’re nicely browned around the edges and somewhat softened. Add the rest of the butter and the scallions and cook until the scallions are soft. Set aside to cool.
In a bowl, whisk the eggs, yolks, and heavy cream until smooth and well blended. Add the mushroom-scallion mixture, the flaked salmon, the minced dill, and the Parmigiano. Season with the salt and pepper.
Heat the oven to 350°F. If using a tart pan with a removable bottom, put it on a baking sheet. Pour the salmon and mushroom mixture into the prepared shell, making sure that the filling is evenly distributed, and bake until the custard is set and the tip of a knife comes out clean and the top is golden brown, 40 to 45 minutes. Let cool for 15 to 20 minutes before serving. Garnish each serving with a sprig of fresh dill.