Heat the olive oil in the skillet. Add the mushrooms, season with salt and pepper and cook over moderate heat until their liquid evaporates, about 4 minutes. Add the shallot and cook, stirring occasionally, until the mushrooms start to brown, about 4 minutes. Fold in the lobster, then transfer the mixture to the prepared baking dish in an even layer. Sprinkle 1 tablespoon of the Parmigiano-Reggiano cheese on top.