Preheat the oven to 350° and lightly butter a 9-by-13-inch baking dish. In a large, deep skillet, melt the 2 tablespoons of butter in the oil. Add the shallots and cook, stirring, for 1 minute. Add the mushrooms, season with salt and pepper and cook over high heat, stirring occasionally, until the mushrooms are browned, about 18 minutes. Add the wine and boil until evaporated, about 1 minute. Add 2 cups of the cream and bring to a boil. Simmer over moderate heat until slightly reduced, about 5 minutes. Stir in the parsley and season with salt and pepper.
Meanwhile, in a large pot of boiling salted water, cook the lasagna sheets until barely al dente. Fill a large bowl with ice water. Drain the sheets and immediately transfer them to the ice water to cool. Drain and pat dry.
Arrange one-fourth of the lasagna sheets in the prepared baking dish. Top with one-third of the mushroom mixture and one-fourth of the Taleggio cubes. Repeat this layering twice more, ending with a layer of pasta. Brush with the remaining 2 tablespoons of cream and scatter the remaining Taleggio cubes on top.
Cover the lasagna very loosely with buttered parchment paper and bake in the upper third of the oven for 15 minutes. Remove the parchment paper and bake for about 15 minutes longer, until the top is golden in spots. Cover loosely and let rest for 15 minutes before serving.