How to Make It
Sprinkle the eggplant slices on both sides with salt and let stand for 1 hour.
Meanwhile, in a small bowl, blend 4 tablespoons of the butter with 1/2 teaspoon of the five-spice powder and season with salt and pepper.
In a medium skillet, melt the remaining 1 1/2 tablespoons of butter. Add the mushrooms, cover and cook over moderate heat, stirring a few times, until browned, about 6 minutes. Stir in the scallions and the remaining 1/4 teaspoon of five-spice powder and season with salt and pepper; let cool.
Light a grill. Gently loosen the skin on the chicken breasts. Spread the mushroom mixture under the skin and smooth out the skin.
Pat the eggplant slices dry with paper towels and brush with oil. Grill over moderate heat until nicely charred and tender, about 5 minutes per side. Keep warm.
Drizzle the chicken breasts with oil and season with salt and pepper. Grill, skin side down, over moderate heat, until lightly browned and crisp, about 10 minutes. Turn and grill until nicely browned, about 10 minutes. Continue grilling, turning a few times, until an instant-read thermometer inserted in the thickest part of the breasts registers 165°, about 5 minutes longer. Transfer the chicken to a carving board and let rest for 5 minutes.
Slice the chicken breasts crosswise 3/4 inch thick. Arrange the chicken and eggplant on plates, dollop with the five-spice butter and serve.