Wrap eggplant in foil and bake in 375 degrees oven until tender, 20 minutes. Melt butter in skillet over medium heat; add mushrooms. Season and cook, stirring, until golden, 7 minutes. Add shallot and sage; cook 2 minutes. Remove from heat; stir in mozzarella. Make lengthwise slit down center of each eggplant. Stuff with mushrooms and pecorino, return to oven and bake until cheese is melted, 7 to 10 minutes. Slice crosswise into pieces.