Preheat the oven to 350°. Butter a 9-by-13-inch baking dish. In a medium skillet, melt the butter. Add the lobster and cook over moderate heat, stirring, until the lobster is hot and the butter turns orange, about 2 minutes. Season with salt and pepper. Using a slotted spoon, transfer the lobster to a plate.
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.
In a medium bowl, beat the eggs. Whisk in the cream, 1/2 teaspoon of salt and 1/4 teaspoon of pepper. Pour the custard over the lobster and mushrooms. Sprinkle the remaining 1 tablespoon of Parmigiano-Reggiano cheese on top and bake for about 25 minutes, until firm. Let the quiche cool to warm.
Cut the quiche into 48 squares. Arrange the pita chips on a platter, set a quiche square on each chip and serve.